TRACT I. A Diſſertation on the Nature and Value of Infinite Series.

1. ABOUT five years ſince I diſcovered a very general and eaſy method of valuing ſeries whoſe terms are alternately poſitive and negative, which equally applies to ſuch ſeries, whether they be converging, or diverging, or their terms all equal; together with ſeve⯑ral other properties relating to certain ſeries: and as we ſhall have occaſion to deliver ſome of thoſe matters in the courſe of theſe tracts, I ſhall take this opportunity of premiſing a few ideas and remarks on the nature and valuation of ſome of the claſſes of ſeries which form the object of thoſe communications. This is done with a view to obviate [2]any miſconceptions that might, perhaps, be made concerning the idea annexed to the term value of ſuch ſeries in thoſe intended tracts, and the ſenſe in which it is there always to be underſtood; which is the more neceſſary, as many controverſies have been warmly agitated concerning theſe matters, not only of late, by ſome of our own countrymen, but alſo by others among the ableſt mathematicians in Europe, at different periods in the courſe of the preſent century; and all this, it ſeems, through the want of ſpecifying in what ſenſe the term value or ſum was to be underſtood in their diſſertations. And in this diſcourſe, I ſhall follow, in a great meaſure, the ſentiments and manner of the late famous L. Euler, contained in a ſimilar memoir of his in the fifth volume of the New Peterſburgh Commentaries, adding and intermix⯑ing here and there other remarks and obſervations of my own.

2. By a converging ſeries, I mean ſuch a one whoſe terms continually decreaſe; and by a diverging ſeries, that whoſe terms continually in⯑creaſe. So that a ſeries whoſe terms neither increaſe nor decreaſe, but are all equal, as they neither converge nor diverge, may be called a neutral ſeries, as a − a + a − a + &c. Now converging ſeries, being ſuppoſed infinitely continued, may have their terms decreaſing to o as a limit, as the ſeries 1 − ½ + ⅓ − ¼ + &c. or only decreaſing to ſome finite magnitude as a limit, as the ſeries 2/1 − 3/2 + 4/3 − 5/4 + &c. which tends continually to 1 as a limit. So in like manner, diverging ſeries may have their terms tending to a limit that is either finite or infinitely great; thus the terms 1 − 2 + 3 − 4 + &c. diverge to infinity, but the diverging terms ½ − ⅔ + ¾ − ⅘ + &c. only to the finite magnitude 1. Hence then, as the ultimate terms of ſeries which do not converge to o, by ſuppoſing them continued in infinitum, may be either finite or infinite, there will be two kinds of ſuch ſeries, each of which will be farther divided into two ſpecies, according as the terms ſhall either be all affected with the ſame ſign, or have alternately the ſigns + and −. We ſhall, therefore, have altogether four ſpecies of ſeries which do not converge to o, an example of each of which may be as here follows: [3]

• I. 1 + 1 + 1 + 1 + 1 + 1 + &c.
• I. ½ + ⅔ + ¾ + ⅘ + ⅚ + 6/7 + &c.
• II. 1 − 1 + 1 − 1 + 1 − 1 + &c.
• II. ½ − ⅔ + ¾ − ⅘ + ⅚ − 6/7 + &c.
• III. 1 + 2 + 3 + 4 + 5 + 6 + &c.
• III. 1 + 2 + 4 + 8 + 16 + 32 + &c.
• IV. 1 − 2 + 3 − 4 + 5 − 6 + &c.
• IV. 1 − 2 + 4 − 8 + 16 − 32 + &c.

3. Now concerning the ſums of theſe ſpecies of ſeries, there have been great diſſenſions among mathematicians; ſome affirming that they can be expreſſed by a certain ſum, while others deny it. In the firſt place, however, it is evident that the ſums of ſuch ſeries as come under the firſt of theſe ſpecies, will be really infinitely great, ſince by actually collecting the terms, we can arrive at a ſum greater than any propoſed number whatever: and hence there can be no doubt but that the ſums of this ſpecies of ſeries may be exhibited by expreſſions of this kind a/0. It is concerning the other ſpecies, therefore, that mathe⯑maticians have chiefly differed; and the arguments which both ſides allege in defence of their opinions, have been endued with ſuch force, that neither party could hitherto be brought to yield to the other.

4. As to the ſecond ſpecies, the famous Leibnitz was one of the firſt who treated of this ſeries 1 − 1 + 1 − 1 + 1 − 1 + &c. and he concluded the ſum of it to = ½, relying upon the following cogent reaſons. And firſt, that this ſeries ariſes by reſolving the fraction 1/1+a into the ſeries 1 − a + a2 − a3 + a4 − a5 + &c. by continual diviſion in the uſual way, and taking the value of a equal to unity. Secondly, for more confirmation, and for perſuading ſuch as are not accuſtomed to cal⯑culations, he reaſons in the following manner: If the ſeries terminate any where, and if the number of the terms be even, then its value will be = 0; [4]but if the number of terms be odd, the value of the ſeries will be = 1: but becauſe the ſeries proceeds in infinitum, and that the number of the terms cannot be reckoned either odd or even, we may conclude that the ſum is neither = 0, nor = 1, but that it muſt obtain a certain middle value, equidifferent from both, and which is therefore = ½. And thus, he adds, nature adheres to the univerſal law of juſtice, giving no partial preference to either ſide.

5. Againſt theſe arguments the adverſe party make uſe of ſuch objec⯑tions as the following. Firſt, that the fraction 1/1+a is not equal to the infinite ſeries 1 − a + a2 − a3 + &c. unleſs a be a fraction leſs than unity. For if the diviſion be any where broken off, and the quotient of the remainder be added, the cauſe of the paralogiſm will be mani⯑feſt; for we ſhall then have [...]; and that although the number n ſhould be made infi⯑nite, yet the ſupplemental fraction [...] ought not to be omitted, unleſs it ſhould become evaneſcent, which happens only in thoſe caſes in which a is leſs than 1, and the terms of the ſeries converge to 0. But that in other caſes there ought always to be included this kind of ſupplement [...]; and although it be affected with the dubious ſign [...], namely − or + according as n ſhall be an even or an odd number, yet if n be infinite, it may not therefore be omitted, under the pretence that an infinite number is neither odd nor even, and that there is no reaſon why the one ſign ſhould be uſed rather than the other; for it is abſurd to ſuppoſe that there can be any integer number, even although it be infinite, which is neither odd nor even.

6. But this objection is rejected by thoſe who attribute determinate ſums to diverging ſeries, becauſe it conſiders an infinite number as a determinate number, and therefore either odd or even, when it is really [5]indeterminate. For that it is contrary to the very idea of a ſeries, ſaid to proceed in infinitum, to conceive any term of it as the laſt, although infinite: and that therefore the objection above-mentioned, of the ſupplement to be added or ſubtracted, naturally falls of itſelf. Therefore, ſince an infinite ſeries never terminates, we never can arrive at the place where that ſupplement muſt be joined; and therefore that the ſupplement not only may, but indeed ought to be neglected, becauſe there is no place found for it.

And theſe arguments, adduced either for or againſt the ſums of ſuch ſeries as above, hold alſo in the fourth ſpecies, which is not otherwiſe embarraſſed with any further doubts peculiar to itſelf.

7. But thoſe who diſpute againſt the ſums of ſuch ſeries, think they have the firmeſt hold in the third ſpecies. For although the terms of theſe ſeries continually increaſe, and that, by actually collect⯑ing the terms, we can arrive at a ſum greater than any aſſignable num⯑ber, which is the very definition of infinity; yet the patrons of the ſums are forced to admit, in this ſpecies, ſeries whoſe ſums are not only finite, but even negative, or leſs than nothing. For ſince the fraction 1/1−a•, by evolving it by diviſion, becomes 1 + a + a2 + a3 + a4 + &c. we ſhould have

• 1/1−2 = − 1 = 1 + 2 + 4 + 8 + 16 + &c.
• 1/1−3 = − ½ = 1 + 3 + 9 + 27 + 81 + &c.

which their adverſaries, not undeſervedly, hold to be abſurd, ſince by the addition of affirmative numbers, we can never obtain a negative ſum; and hence they urge that there is the greater neceſſity for includ⯑ing the before-mentioned ſupplement additive, ſince by taking it in, it is evident that − 1 = 1 + 2 + 4 + 8 ........ 2ⁿ + 2ⁿ+1/1−2, although n ſhould be an infinite number.

[6]8. The defenders therefore of the ſums of ſuch ſeries, in order to reconcile this ſtriking paradox, more ſubtle perhaps than true, make a diſtinction between negative quantities; for they argue that while ſome are leſs than nothing, there are others greater than infinite, or above infinity. Namely, that the one value of −1 ought to be underſtood, when it is conceived to ariſe from the ſubtraction of a greater number a + 1 from a leſs a; but the other value, when it is found equal to the ſeries 1 + 2 + 4 + 8 + &c. and ariſing from the diviſion of the number 1 by −1; for that in the former caſe it is leſs than nothing, but in the latter greater than infinite. For the more confirmation, they bring this example of fractions ¼, ⅓, ½, 1/1, 1/0, 1/−1, 1/−2, 1/−3, &c. which, evidently increaſing in the leading terms, it is inferred will con⯑tinually increaſe; and hence they conclude that 1/−1 is greater than 1/0, and 1/−2 greater than 1/−1, and ſo on: and therefore as 1/−1 is expreſſed by −1, and 1/0 by ∼ or infinity, −1 will be greater than ∼, and much more will −½ be greater than ∼. And thus they ingeniouſly enough repel⯑led that apparent abſurdity by itſelf.

9. But although this diſtinction ſeemed to be ingeniouſly deviſed, it gave but little ſatisfaction to the adverſaries; and beſides, it ſeemed to affect the truth of the rules of algebra. For if the two values of −1, namely 1 − 2 and 1/−1, be really different from each other, as we may not confound them, the certainty and the uſe of the rules, which we follow in making calculations, would be quite done away; which would be a greater abſurdity than that for whoſe ſake the diſtinction was deviſed: but if 1 − 2 = 1/−1, as the rules of algebra require, for by multiplication [...], the matter in debate is not ſettled; ſince the quantity −1, to which the ſeries 1 + 2 + 4 + 8 + &c. is made equal, is leſs than nothing, and therefore the ſame difficulty ſtill remains. In the mean time however, it ſeems but agreeable to truth, to ſay that the ſame quantities which are below nothing, may be [7]taken as above infinite. For we know, not only from algebra, but from geometry alſo, that there are two ways, by which quantities paſs from poſitive to negative, the one through the cypher or nothing, and the other through infinity: and beſides that quantities, either by in⯑creaſing or decreaſing from the cypher, return again, and revert to the ſame term o; ſo that quantities more than infinite are the ſame with quantities leſs than nothing, like as quantities leſs than infinite agree with quantities greater than nothing.

10. But, farther, thoſe who deny the truth of the ſums that have been aſſigned to diverging ſeries, not only omit to aſſign other values for the ſums, but even ſet themſelves utterly to oppoſe all ſums belong⯑ing to ſuch ſeries, as things merely imaginary. For a converging ſeries, as ſuppoſe this 1 + ½ + ¼ + ⅛ + &c. will admit of a ſum = 2, becauſe the more terms of this ſeries we actually add, the nearer we come to the number 2: but in diverging ſeries the caſe is quite different; for the more terms we add, the more do the ſums which are produced differ from one another, neither do they ever tend to any certain deter⯑minate value. Hence they conclude that no idea of a ſum can be applied to diverging ſeries, and that the labour of thoſe perſons who employ themſelves in inveſtigating the ſums of ſuch ſeries, is manifeſtly uſeleſs, and indeed contrary to the very principles of analyſis.

11. But notwithſtanding this ſeemingly real difference, yet neither party could ever convict the other of any error, whenever the uſe of ſeries of this kind has occurred in analyſis; and for this good reaſon, that neither party is in an error, but that the whole difference conſiſts in words only. For if in any calculation I arrive at this ſeries 1 − 1 + 1 − 1 + &c. and that I ſubſtitute ½ inſtead of it; I ſhall ſurely not thereby commit any error; which however I ſhould certainly incur if I ſubſtitute any other number inſtead of that ſeries; and hence there remains no doubt but that the ſeries 1 − 1 + 1 − 1 + &c. and the [8]fraction ½, are equivalent quantities, and that the one may always be ſubſtituted inſtead of the other without error. So that the whole mat⯑ter in diſpute ſeems to be reduced to this only, namely, whether the fraction ½ can be properly called the ſum of the ſeries 1 − 1 + 1 − 1 + &c. Now if any perſons ſhould obſtinately deny this, ſince they will not however venture to deny the fraction to be equivalent to the ſeries, it is greatly to be feared they will fall into mere quarrelling about words.

12. But perhaps the whole diſpute will eaſily be compromiſed, by carefully attending to what follows. Whenever, in analyſis, we arrive at a complex function or expreſſion, either fractional or tranſcendental; it is uſual to convert it into a convenient ſeries, to which the remaining calculus may be more eaſily applied. And hence the occaſion and riſe of infinite ſeries. So far only then do infinite ſeries take place in analytics, as they ariſe from the evolution of ſome finite expreſſion; and therefore, inſtead of an infinite ſeries, in any calculus, we may ſubſtitute that formula, from whoſe evolution it aroſe. And hence, for performing calculations with more eaſe or more benefit, like as rules are uſually given for converting into infinite ſeries ſuch finite expreſſions as are endued with leſs proper forms; ſo, on the other hand, thoſe rules are to be eſteemed not leſs uſeful by the help of which we may inveſti⯑gate the finite expreſſion from which a propoſed infinite ſeries would reſult, if that finite expreſſion ſhould be evolved by the proper rules: and ſince this expreſſion may always, without error, be ſubſtituted in⯑ſtead of the infinite ſeries, they muſt neceſſarily be of the ſame value: and hence no infinite ſeries can be propoſed, but a finite expreſſion may, at the ſame time, be conceived as equivalent to it.

13. If therefore, we only ſo far change the received notion of a ſum as to ſay, that the ſum of any ſeries, is the finite expreſſion by the evolution of which that ſeries may be produced, all the difficulties, [9]which have been agitated on both ſides, vaniſh of themſelves. For, firſt, that expreſſion by whoſe evolution a converging ſeries is produced, exhibits at the ſame time its ſum, in the common acceptation of the term: neither, if the ſeries ſhould be divergent, could the inveſtiga⯑tion be deemed at all more abſurd, or leſs proper, namely, the ſearch⯑ing out a finite expreſſion which, being evolved according to the rules of algebra, ſhall produce that ſeries. And ſince that expreſſion may be ſubſtituted in the calculation inſtead of this ſeries, there can be no doubt but that it is equal to it. Which being the caſe, we need not neceſſarily deviate from the uſual mode of ſpeaking, but might be per⯑mitted to call that expreſſion alſo the ſum, which is equal to any ſeries whatever, provided however, that, in ſeries whoſe terms do not con⯑verge to o, we do not connect that notion with this idea of a ſum, namely, that the more terms of the ſeries are actually collected, the nearer we muſt approach to the value of the ſum.

14. But if any perſon ſhall ſtill think it improper to apply the term ſum, to the finite expreſſions by whoſe evolution all ſeries in general are produced; it will make no difference in the nature of the thing; and inſtead of the word ſum, for ſuch finite expreſſion, he may uſe the term value; or perhaps the term radix would be as proper as any other that could be employed for this purpoſe, as the ſeries may juſtly be conſidered as iſſuing or growing out of it, like as a plant ſprings from its root, or from its ſeed. The choice of terms being in a great mea⯑ſure arbitrary, every perſon is at liberty to employ them in whatever ſenſe he may think fit, or proper for the purpoſe in hand; provided always that he fix and determine the ſenſe in which he underſtands or employs them. And as I conſider any ſeries, and the finite expreſſion by whoſe evolution that ſeries may be produced, as no more than two different ways of expreſſing one and the ſame thing, whether that finite expreſſion be called the ſum, or value, or radix of the ſeries; ſo in the following paper, and in ſome others which may perhaps hereafter [10]be produced, it is in this ſenſe I deſire to be underſtood when ſearching out the value of ſeries, namely, that the object of my enquiry, is the radix by whoſe evolution the ſeries may be produced, or elſe an ap⯑proximation to the value of it in decimal numbers, &c.

Royal Military Acad. Woolwich, May 24, 1785.

TRACT II. A new Method for the Valuation of Numeral Infinite Series, whoſe Terms are alternately (+) Plus and (−) Minus; by taking continual Arith⯑metical Means between the Succeſſive Sums, and their Means.

Article 1. THE remarkable difference between the facility which mathematicians have found in their endeavours to determine the values of infinite ſeries whoſe terms are alternately affirmative and negative, and the difficulty of doing the ſame thing with reſpect to thoſe ſeries whoſe terms are all affirmative, is one of thoſe ſtriking appearances in ſcience which we can hardly perſuade our⯑ſelves is true, even after we have ſeen many proofs of it, and which ſerve to put us ever after on our guard not to truſt to our firſt notions, or conjectures, on theſe ſubjects, till we have brought them to the teſt of demonſtration. For, at firſt ſight it is very natural to imagine, that thoſe infinite ſeries whoſe terms are all affirmative, or added to the firſt term, muſt be much ſimpler in their nature, and much eaſier to be ſummed, than thoſe whoſe terms are alternately affirmative and ne⯑gative; which, nevertheleſs, we find, upon examination, to be directly contrary to the truth; the methods of finding the ſums of the latter ſeries being numerous and eaſy, and alſo very general, whereas thoſe that have been hitherto diſcovered for the ſummation of the former ſeries, are few and difficult, and confined to ſeries whoſe terms are generated from each other according to ſome particular laws, inſtead of extending, as the other methods do, to all ſorts of ſeries, whoſe terms are connected together by addition, by whatever law their terms are formed. Of this remarkable difference between theſe two ſorts of ſeries, the new method of finding the ſums of thoſe whoſe terms are [12]alternately poſitive and negative, which is the ſubject of the preſent paper, will afford us a ſtriking inſtance, as it poſſeſſes the happy qua⯑lities of ſimplicity, eaſe, perſpicuity, and univerſality in a great degree; and yet, as the eſſence of it conſiſts in the alternation of the ſigns + and −, by which the terms are connected with the firſt term, it is of no uſe in the ſummation of thoſe other ſeries whoſe terms are all con⯑nected with each other by the ſign +.

2. This method, ſo eaſy and general, is, in ſhort, ſimply this: be⯑ginning at the firſt term a of the ſeries a − b + c − d + e − f + &c. which is to be ſummed, compute ſeveral ſucceſſive values of it, by taking in ſucceſſively more and more terms, one term being taken in at a time; ſo that the firſt value of the ſeries ſhall be its firſt term a, (or even o or nothing may begin the ſeries of ſums); the next value ſhall be its firſt two terms a − b, reduced to one number; its next value ſhall be the firſt three terms a − b + c, reduced to one number; its next value ſhall be the firſt four terms a − b + c − d, reduced alſo to one number; and ſo on. This, it is evident, may be done by means of the eaſy arithmetical operation of addition and ſubtraction. And then, having found a ſufficient number of ſucceſſive values of the ſeries, more or leſs as the caſe may require, interpoſe between theſe values a ſet of arithmetical mean quantities or proportionals; and between theſe arithmetical means interpoſe a ſecond ſet of arithmetical mean quanti⯑ties; and between thoſe arithmetical means of the ſecond ſet, inter⯑poſe a third ſet of arithmetical mean quantities; and ſo on as far as you pleaſe. By this proceſs we ſoon find either the true value of the ſeries propoſed, when it has a determinate rational value, or otherwiſe we obtain ſeveral ſets of values approximating nearer and nearer to the ſum of the ſeries, both in the columns and in the lines, either horizon⯑tal or obliquely deſcending or aſcending; namely, both of the ſeveral ſets of means themſelves, and the ſets or ſeries formed of any of their correſponding terms, as of all their firſt terms, of their ſecond terms, [13]of their third terms, &c. or of their laſt terms, of their penultimate terms, of their antepenultimate terms, &c. and if between any of theſe latter ſets, conſiſting of the like or correſponding terms of the former ſets of arithmetical means, we again interpoſe new ſets of arithmetical means, as we did at firſt with the ſucceſſive ſums, we ſhall obtain other ſets of approximating terms, having the ſame properties as the former. And thus we may repeat the proceſs as often as we pleaſe, which will be found very uſeful in the more difficult diverging ſeries, as we ſhall ſee hereafter. For this method, being derived only from the circum⯑ſtance of the alternation of the ſigns of the terms (+ and −), it is there⯑fore not confined to converging ſeries alone, but is equally applicable both to diverging ſeries, and to neutral ſeries, (by which laſt name I ſhall take the liberty to diſtinguiſh thoſe ſeries whoſe terms are all of the ſame conſtant magnitude); namely, the application is equally the ſame for all the three following ſorts of ſeries, viz.

• Converging, 1 − ½ + ⅓ − ¼ + ⅕ − ⅙ + &c.
• Diverging, 1 − 2 + 3 − 4 + 5 − 6 + &c.
• Neutral, 1 − 1 + 1 − 1 + 1 − 1 + &c.

As is demonſtrated in what follows, and exemplified in a variety of inſtances.

It muſt be noted, however, that by the value of the ſeries, I always mean ſuch radix, or finite expreſſion, as, by evolution, would produce the ſeries in queſtion; according to the ſenſe I have ſtated in the former paper, on this ſubject; or an approximate value of ſuch radix; and which radix, as it may be ſubſtituted inſtead of the ſeries in any operation, I call the value of the ſeries.

3. It is an obvious and well-known property of infinite ſeries, with alternate ſigns, that when we ſeek their value by collecting their terms one after another, we obtain a ſeries of ſucceſſive ſums, which approach continually nearer and nearer to the true value of the propoſed ſeries, when it is a converging one, or one whoſe terms always decreaſe [14]by ſome regular law; but in a diverging ſeries, or one whoſe terms as continually increaſe, thoſe ſucceſſive ſums diverge always more and more from the true value of the ſeries. And from the circumſtance of the alternate change of the ſigns, it is alſo a property of thoſe ſuc⯑ceſſive ſums, that when the laſt term which is included in the collection, is a poſitive one, then the ſum obtained is too great, or exceeds the truth; but when the laſt collected term is negative, then the ſum is too little, or below the truth. So that, in both the converging and diverging ſeries, the firſt term alone, being poſitive, exceeds the truth; the ſecond ſum, or the ſum of the firſt two terms, is below the truth; the third ſum, or the ſum of three terms, is above the truth; the fourth ſum, or the ſum of four terms, is below the truth; and ſo on; the ſum of any even number of terms being below the true value of the ſeries, and the ſum of any odd number, above it. All which is generally known, and evident from the nature and form of the ſeries. So, of the ſeries a − b + c − d + e − f + &c. the firſt ſum a is too great; the ſecond ſum a − b too little; the third ſum a − b + c too great; and ſo on as in the following table, where s is the true value of the ſeries, and o is placed before the collected ſums, to compleat the ſeries, being the value when no terms are included:

4. Hence the value of every alternate ſeries s, is poſitive, and leſs than the firſt term a, the ſeries being always ſuppoſed to begin with a poſitive term a; and conſequently if the ſigns of all the terms be [15]changed, or if the ſeries begin with a negative term, the value s will ſtill be the ſame, but negative, or the ſign of the ſum will be changed, and the value become −s = −a + b − c + d − &c. Alſo, becauſe the ſucceſſive ſums, in a converging ſeries, always ap⯑proach nearer and nearer to the true value, while they recede always farther and farther from it in a diverging ſeries; it follows that, in a neutral ſeries, a − a + a − a + &c. which holds a middle place between the two former, the ſucceſſive ſums o, a, o, a, o, a &c. will neither converge nor diverge, but will be always at the ſame diſ⯑tance from the value of the propoſed ſeries a − a + a − a + &c. and conſequently that value will always be = ½a, which holds every where the middle place between o and a.

5. Now, with reſpect to a converging ſeries, a − b + c − d + &c. becauſe o is below, and a above s the value of the ſeries, but a nearer than o to the value s, it follows that s lies between a and ½a, and that ½a is leſs than s, and ſo nearer to s than o is. In like manner, be⯑cauſe a is above, and a − b below the value s, but a − b nearer to that value than a is, it follows that s lies between a and a − b, and that the arithmetical mean a − ½b is ſomething above the value of s, but nearer to that value than a is. And thus, the ſame reaſoning holding in every following pair of ſucceſſive ſums, the arithmetical means between them will form another ſeries of terms, which are, like thoſe ſums, alternately leſs and greater than the value of the pro⯑poſed ſeries, but approximating nearer to that value than the ſeveral ſucceſſive ſums do, as every term of thoſe means is nearer to the value s than the correſponding preceding term in the ſums is. And like as the ſucceſſive ſums form a progreſſion approaching always nearer and nearer to the value of the ſeries, ſo in like manner their arithmetical means form another progreſſion coming nearer and nearer to the ſame value, and each term of the progreſſion of means nearer than each term of the ſucceſſive ſums. Hence then we have the two following [16]ſeries, namely, of ſucceſſive ſums and their arithmetical means, in which each ſtep approaches nearer to the value of s than the former, the latter progreſſion being however nearer than the former, and the terms or ſteps of each alternately below and above the value s of the ſeries a − b + c − d + &c.

where the mark ›, placed before any ſtep, ſignifies that it is too little, or below the value s of the converging ſeries a − b + c − d + &c. and the mark ‹ ſignifies the contrary, or too great. And hence ½a, or half the firſt term of ſuch a converging ſeries, is leſs than s the value of the ſeries.

6. And ſince theſe two progreſſions poſſeſs the ſame properties, but only the terms of the latter nearer to the truth than the former; for the very ſame reaſons as before, the means between the terms of theſe firſt arithmetical means, will form a third progreſſion, whoſe terms will approach ſtill nearer to the value of s than the ſecond progreſſion, or the firſt means; and the means of theſe ſecond means will approach nearer than the ſaid ſecond means do; and ſo on continually, every ſucceeding order of arithmetical means, approaching nearer to the value of s than the former. So that the following columns of ſums and means will be each nearer to the value of s than the former, viz. [17]

Where every column conſiſts of a ſet of quantities, approaching ſtill nearer and nearer to the value of s, the terms of each column being alternately below and above that value, and each ſucceeding column approaching nearer than the preceding one. Alſo every line, formed of all the firſt terms, all the ſecond terms, all the third terms, &c. of the columns, forms alſo a progreſſion whoſe terms continually approximate to the value of s, and each line nearer or quicker than the former; but differing from the columns, or vertical progreſſions, in this, namely, that whereas the terms in the columns are alternately below and above the value of s, thoſe in each line are all on one ſide of the value s, namely, either all below or all above it; and the lines alternately too little and too great, namely all the expreſſions in the firſt line too little, all thoſe in the ſecond line too great, thoſe in the third line too little, and ſo on, every odd line being too little, and every even line too great.

7. Hence the expreſſions 0, a/2, 3a−b/4, 7a−4b+c/8, 15a−11b+5c−a/16, 31a−26b+16c−6d+e/32, &c. are continual approximations to the value s of the converging ſeries a − b + c − d + e − &c. and are all below the truth. But we can eaſily expreſs all theſe ſeveral theorems by one general formula. For, ſince it is evident by the conſtruction, that whilſt the denominator in any one of them is ſome power (2ⁿ) of 2 or 1 + 1, the numeral coefficients [18]of a, b, c, &c. the terms in the numerator, are found by ſubtracting all the terms except the laſt term, one after another, from the ſaid power 2ⁿ or [...] which is = 1 + n + n · n−1/2 + n · n−2/3 + &c. namely the coefficient of a equal to all the terms 2ⁿ minus the firſt term 1; that of b equal to all except the firſt two terms 1 + n; that of c equal to all except the firſt three; and ſo on, till the coeffi⯑cient of the laſt term be = 1 the laſt term of the power; it follows that the general expreſſion for the ſeveral theorems, or the general ap⯑proximate value of the converging ſeries a − b + c − d + &c. will be [...] continued till the terms vaniſh and the ſeries break off, n being equal to o or any integer number. Or this general formula may be expreſſed by this ſeries, [...] where A, B, C, &c. denote the coefficients of the ſeveral preceding terms. And this expreſſion, which is always too little, is the nearer to the true value of the ſeries a − b + c − d + &c. as the number n is taken greater: always excepting however thoſe caſes in which the theorem is accurately true when n is ſome certain finite number. Alſo, with any value of n, the formula is nearer to the truth, as the terms a, b, c, &c. of the propoſed ſeries, are nearer to equality; ſo that the ſlower the propoſed ſeries converges, the more accurate is the formula, and the ſooner does it afford a near value of that ſeries: which is a very fa⯑vourable circumſtance, as it is in caſes of very ſlow convergency that approximating formulae are chiefly wanted. And, like as the formula approaches nearer to the truth as the terms of the ſeries approach to an equality, ſo when the terms become quite equal, as in a neutral ſeries, the formula becomes quite accurate, and always gives the ſame value ½a for s or the ſeries, whatever integer number be taken for n. And farther, when the propoſed ſeries, from being converging, paſſes through [19]neutrality, when its terms are equal, and becomes diverging, the for⯑mula will ſtill hold good, only it will then be alternately too great, and too little as long as the ſeries diverges, as we ſhall preſently ſhew more fully. So that in general the value s of the ſeries a − b + c − d + &c. whether it be converging, diverging, or neutral, is leſs than the firſt term a; when the ſeries converges, the value is above ½a; when it diverges, it is below ½a; and when neutral, it is equal to ½a.

8. Take now the ſeries of the firſt terms of the ſeveral orders of arithmetical means, which form the progreſſion of continual approxi⯑mating formulae, being each nearer to the value of the ſeries a − b + c − d + &c. than the former, and place them in a column one under another; then take the differences between every two adjacent formulae, and place in another column by the ſide of the former, as here below:

From which it appears, that this ſeries of differences conſiſts of the very ſame quantities, which form the firſt terms of all the orders of differ⯑ences of the terms of the propoſed ſeries a − b + c − d + &c. when taken as uſual in the differential method. And becauſe the firſt of the above differences added to the firſt formula, gives the ſecond formula; and the ſecond difference added to the ſecond formula, gives the third formula; and ſo on; therefore the firſt formula with all the differences added, will give the laſt formula; conſequently our general formula [...] [20]which approaches to the value of the ſeries a − b + c − d + &c. is alſo equivalent to, or reduces to this form, [...] which, it is evident, agrees with the famous differential ſeries. And this coincidence might be ſufficient to eſtabliſh the truth of our method, though we had not given other more direct proof of it. Hence it ap⯑pears then, that our theorem is of the ſame degree of accuracy, or convergency, as the differential theorem; but admits of more direct and eaſy application, as the terms themſelves are uſed, without the pre⯑vious trouble of taking the ſeveral orders of differences. And our method will be rendered general for literal as well as numeral ſeries, by ſuppoſing a, b, c, &c. to repreſent, not barely the coefficients of the terms, but the whole terms, both the numeral and the literal part of them. However, as the chief uſe of my method is to obtain the nu⯑meral value of ſeries, when a literal ſeries is to be ſo ſummed, it is to be made numeral by ſubſtituting the numeral values of the letters in⯑ſtead of them. It is farther evident, that we might eaſily derive our method of arithmetical means from the above differential ſeries, by beginning with it, and receding back to our theorems, by a counter proceſs to that above given.

9. Having, in Art. 5, 6, 7, 8, compleated the inveſtigations for the firſt or converging form of ſeries, the firſt four articles being introduc⯑tory to both forms in common; we may now proceed to the diverging form of ſeries, for which we ſhall find the ſame method of arithmetical means, and the ſame general formula, as for the converging ſeries; as well as the mode of inveſtigation uſed in Art. 5 et ſeq. only changing ſometimes greater for leſs, or leſs for greater. Thus then, reaſoning from the table of ſucceſſive ſums in Art. 3, in which s is alternately above and below the expreſſions o, a, a − b, a − b + c, &c. be⯑cauſe o is below, and a above the value s of the ſeries a − b + c − d + &c. but o nearer than a to that value, it follows that s lies be⯑tween [21]o and ½a, and that ½a is greater than s, but nearer to s than a is. In like manner, becauſe a is above, and a − b below the value s, but a nearer to that value than a − b is, it follows that s lies between a and a − b, and that the arithmetical mean a − ½b is below s, but that it is nearer to s than a − b is. And thus, the ſame reaſoning holding in every pair of ſucceſſive ſums, the arithmetical means between them will form another ſeries of terms, which are alternately greater and leſs than s the value of the propoſed ſeries; but here greater and leſs in the con⯑trary way to what they were for the converging ſeries, namely, thoſe ſteps greater here which were leſs there, and leſs here which before were greater. And this firſt ſet of arithmetical means, will either con⯑verge to the value of s, or will at leaſt diverge leſs from it than the progreſſion of ſucceſſive ſums. Again, the ſame reaſoning ſtill hold⯑ing good, by taking the arithmetical means of thoſe firſt means, another ſet is found, which will either converge, or elſe diverge leſs than the former. And ſo on as far as we pleaſe, every new operation gradually checking the firſt or greateſt divergency, till a number of the firſt terms of a ſet converge ſufficiently faſt, to afford a near value of s the pro⯑poſed ſeries.

10. Or, by taking the firſt terms of all the orders of means, we find the ſame ſet of theorems, namely [...], &c. or in general, [...] which will be alternately above and below s the value of the ſeries, till the divergency is overcome. Then this ſeries, which conſiſts of the firſt terms of the ſeveral orders of means, may be treated as the ſucceſſive ſums, taking ſeveral orders of means of theſe again. After which the firſt terms of theſe laſt orders may be treated again in the ſame manner; and ſo on as far as we pleaſe. Or the ſeries of ſecond terms, or third terms, &c. or ſometimes, the terms aſcending ob⯑liquely, may be treated in the ſame manner to advantage. And [22]with a little practice and inſpection of the ſeveral ſeries, whether vertical, or horizontal, or oblique, (for they all tend to the detection of the ſame value s) we ſhall ſoon learn to diſtinguiſh whereabouts the required quantity s is, and which of the ſeries will ſooneſt approxi⯑mate to it.

11. To exemplify now this method, we ſhall take a few ſeries of both ſorts, and find their value ſometimes by actually going through the operations of taking the ſeveral orders of arithmetical means, and at other times by uſing ſome one of the theorems [...], &c. at once. And to render the uſe of theſe theorems ſtill eaſier, we ſhall here ſub⯑join the following table, where the firſt line conſiſting of the powers of 2, contains the denominators of the theorems in their order, and the figures in their perpendicular columns below them, are the coefficients of the ſeveral terms in the numerators of the theorems, namely, the upper figure, next below the power of 2, the coefficient of a; the next below, that of b; the third that of c, &c.

[23]The conſtruction and continuation of this table, is a buſineſs of little labour. For the numbers in the firſt horizontal line next below the line of the powers of 2, are thoſe powers diminiſhed each by unity. The numbers in the next horizontal line, are made from the numbers in the firſt, by ſubtracting from each the index of that power of 2 which ſtands above it. And for the reſt of the table, the formation of it is obvious from this principle, which reigns through the whole, that every number in it is the ſum of two others, namely of the next to it on the left in the ſame horizontal line, and the next above that in the ſame vertical column. So that the whole table is formed from a few of its initial numbers, by eaſy operations of addition.

In converging ſeries, it will be farther uſeful, firſt to collect a few of the initial terms into one ſum, and then apply our method to the fol⯑lowing terms, which will be ſooner valued becauſe they will converge ſlower.

12. For the firſt example, let us take the very ſlowly converging ſeries 1 − ½ + ⅓ − ¼ + ⅕ − ⅙ + &c. which is known to expreſs the hyp. log. of 2, which is = .69314718.

Here a = 1, b = ½, c = ⅓, d = ¼, &c. and the value, as found by theorem the 1ſt, 2d, 3d, 4th, 10th, and 20th, will be thus:

• 1ſt, a/2 = ½ = .5.
• 2d, [...].
• 3d, [...].
• 4th, [...].
• 10th, [...].
• 20th, [...].

Where it is evident that every theorem gives always a nearer value than the former: the 10th theorem gives the value true to the 4th [24]figure, and the 20th theorem to the 8th figure. The operation for the 10th and 20th theorems, will be eaſily performed by dividing, mentally, the numbers in their reſpective columns in the table of coefficients in Art. 11, by the ordinate numbers 1, 2, 3, 4, 5, 6, &c. placing the quotients of the alternate terms below each other, then adding each up, and dividing the difference of the ſums continually five or ten times ſucceſſively by the number 4: after the manner as here placed below, where the operation is ſet down for both of them.

1. For the 10th Theorem. [...]

2. For the 20th Theorem. [...]

Again, to perform the operation by taking the ſucceſſive ſums, and the arithmetical means: let the terms ½, ⅓, ¼, &c. be reduced to decimal numbers, by dividing the common numerator 1 by the deno⯑minators 2, 3, 4, &c. or rather by taking theſe out of the table at the end of my Miſcellanea Mathematica, publiſhed in 1775, which con⯑tains a table of the ſquare roots and reciprocals of all the numbers, [25]1, 2, 3, 4, 5, 6, &c. to 1000, and which is of great uſe in ſuch calculations as theſe. Then the operation will ſtand thus: [...] Here, after collecting the firſt twelve terms, I begin at the bottom, and, aſcending upwards, take a very few arithmetical means between the ſucceſſive ſums, placing them on the right of them: it being unne⯑ceſſary to take the means of the whole, as any part of them will do the buſineſs, but the lower ones in a converging ſeries beſt, becauſe they are nearer the value ſought, and approach ſooner to it. I then take the means of the firſt means, and the means of theſe again, and ſo on, till the value is obtained as near as may be neceſſary. In this proceſs we ſoon diſtinguiſh whereabouts the value lies, the limits or means, which are alternately above and below it, gradually contracting, and approaching towards each other. And when the means are reduced to a ſingle one, and it is found neceſſary to get the value more exactly, I then go back to the columns of ſucceſſive ſums, and find another firſt mean, either next below or above thoſe before found, and continue it through the 2d, 3d, &c. means, which makes now a duplicate in the laſt column of means, and the mean between them gives another ſingle mean of the next order; and ſo on as far as we ſee it neceſſary. By ſuch a gradual progreſs we uſe no more terms nor labour than is quite requiſite for the degree of accuracy required.

Or, after having collected the ſum of any number of terms, we may apply any of the formulae to the following terms. So, having as above [26]found .653211 for the ſum of the firſt 12 terms, and calling the next or 13th term .076923 = a, the 14th term .0714285 = b, the next, .06666 &c. = c, and ſo on: then the 2d theorem 3a−b/4 gives .039835, which added to .653211 the ſum of the firſt 12 terms, gives .693046, the value of the ſeries true in three places of figures; and the 3d theorem [...] gives .039927 for the following terms, and which added to .653211 the ſum of the firſt 12 terms, gives .693138, the value of the ſeries true in five places. And ſo on.

13. For a ſecond example, let us take the ſlowly converging ſeries 2/1 − 3/2 + 4/3 − 5/4 + 6/5 − 7/6 + &c. which is = ½ + hyp. log. of 2 = 1.19314718. Then [...]

Here, after the 3d column of means, the firſt four figures 1.193, which are common, are omitted, to ſave room and the trouble of writing them ſo often down; and in the laſt three columns, the proceſs is repeated with the laſt three figures of each number; and the laſt of theſe 147, joined to the firſt four, give 1.193147 for the value of the ſeries propoſed. And the ſame value is alſo obtained by the theorems uſed as in the former example.

14. For the third example let us take the converging ſeries 1 − ⅓ + ⅕ − 1/7 + 1/9 − 1/11 + &c. which is = .7853981 &c. or ¼ of the circumference of the circle whoſe diameter is 1. Here a = 1, b = ⅓, [27] c = ⅕, &c. then turning the terms into decimals, and proceeding with the ſucceſſive ſums and means as before, we obtain the 5th mean true within a unit in the 6th place as here below: [...]

15. To find the value of the converging ſeries [...] which occurs in the expreſſion for determining the time of a body's deſcent down the arc of a circle:

The firſt terms of this ſeries I find ready computed by Mr. Baron Maſeres, pa. 219 Philoſ. Tranſ. 1777; theſe being arranged under one another, and the ſums collected, &c. as before, give .834625 for the value of that ſeries, being only 1 too little in the laſt figure.

[...]

16. To find the value of 1 − ¼ + ½ − 1/16 + 1/25 − &c. con⯑ſiſting of the reciprocals of the natural ſeries of ſquare numbers. [28] [...] The laſt mean .822467 is true in the laſt figure, the more accurate value of the ſeries 1 − ¼ + 1/9 − 1/16 + &c. being .8224670 &c.

17. Let the diverging ſeries ½ − ⅔ + ¾ − ⅘ + &c. be propoſed; where the terms are the reciprocals of thoſe in Art. 13.

[...]

Here the ſucceſſive ſums are alternately + and −, as well as the terms themſelves of the propoſed ſeries, but all the arithmetical means are poſitive. The numbers in each column of means are alter⯑nately too great and too little, but ſo as viſibly to approach towards each other. The ſame mutual approximation is viſible in all the oblique lines from left to right, ſo that there is a general and mutual tendency, in all the columns, and in all the lines, to the limit of the value of the ſeries. But with this difference, that all the numbers in any line de⯑ſcending obliquely from left to right, are on one ſide of the limit, and [29]thoſe in the next line in the ſame direction, all on the other ſide, the one line having its numbers all too great, while thoſe in the next line are all too little; but, on the contrary, the lines which aſcend from below obliquely towards the right, have their numbers alternately too great and too little, after the manner of thoſe in the columns, but ap⯑proximating quicker than thoſe in the columns. So that, after having continued the columns of arithmetical means to any convenient extent, we may then ſelect the terms in the laſt, or any other line obliquely aſcending from left to right, or rather beginning with the laſt found mean on the right, and deſcending towards the left; then arrange thoſe terms below one another in a column, and take their continual arithme⯑tical means, like as was done with the firſt ſucceſſive ſums, to ſuch extent as the caſe may require. And if neither theſe new columns, nor the oblique lines approach near enough to each other, a new ſet may be formed from one of their oblique lines which has its terms alternately too great and too little. And thus we may proceed as far as we pleaſe. Theſe repetitions will be more neceſſary in treating ſeries which diverge more; and having here once for all deſcribed the properties attending the ſeries, with the method of repetition, we ſhall only have to refer to them as occaſion ſhall offer. In the preſent inſtance, the laſt two or three means vary or differ ſo little, that the limit may be concluded to lie nearly in the middle between them, and therefore the mean between the two laſt 144 and 150, namely 147, may be concluded to be very near the truth, in the laſt three figures; for as to the firſt three figures 193, I dropt the repetition of them after the firſt three columns of means, both to ſave ſpace, and the trouble of writing them ſo often over again. So that the value of the ſeries in queſtion may be concluded to be .193147 very nearly, which is = − ½ + the hyp. log. of 2; or 1 leſs than its reciprocal ſeries in Art. 13.

18. Take the diverging ſeries 5/4 − 5·7/4·6 + 5·7·9/4·6·8 − 5·7·9·11/4·6·8·10 + &c. Here, firſt uſing ſome of the formulae, we have by the

• 1ſt, a/2 = .625
• 2d, [...]
• 3d, [...]
• 4th, [...]
• 5th, [...]. &c.

Or, thus, taking the ſeveral orders of means, &c.

[...]

Here the ſucceſſive ſums are alternately + and −, but the arithme⯑tical means are all +. After the ſecond column of means, the firſt two figures 56 are omitted, being common; and in the laſt three columns the firſt three figures 569, which are common, are omitted. Towards the end, all the numbers, both oblique and vertical, approach ſo near together, that we may conclude that the laſt three figures 035 are all true; and theſe being joined to the firſt three 569, we have .569035 for the value of the ſeries, which is otherwiſe found 2+√2/6 = .56903559 &c.

19. Let us take the diverging ſeries 2²/1 − 3²/2 + 4²/3 − 5²/4 + &c. or 4/1 − 9/2 + 16/3 − 25/4 + &c. or 4 − 4½ + 5⅓ − 6¼ + 7⅕ − 8⅙ + &c.

[31] [...]

After the ſecond column of means, the firſt four figures 1.943 are omitted, being common to all the following columns; to theſe annexing the laſt three figures 147 of the laſt mean, we have 1.943147 for the ſum of the ſeries, which we otherwiſe know is equal to 5/4 + hyp. log. of 2. See Simp. Diſſert. Ex. 2. p. 75 and 76.

And the ſame value might be obtained by means of the formulae, uſing them as before.

20. Taking the diverging ſeries 1 − 2 + 3 − 4 + 5 − &c. the method of means gives us, [...]

Where the ſecond, and every ſucceeding column of means, gives ¼ for the value of the ſeries propoſed.

In like manner, uſing the theorems, the firſt gives ½, but the ſecond, third, fourth, &c. give each of them the ſame value ¼; thus:

• a/2 = ½
• [...]
• [...]
• [...]. &c.

[32]21. Taking the ſeries 1 − 4 + 9 − 16 + 25 − 36 + &c. whoſe terms conſiſt of the ſquares of the natural ſeries of numbers, we have, by the arithmetical means, [...]

Where it is only in the ſecond column of means that the divergency is counteracted; after that the third and all the other orders of means give o for the value of the ſeries 1 − 4 + 9 − 16 + &c.

The ſame thing takes place on uſing the formulae, for

• a/2 = ½
• [...]
• [...]
• [...]

where the third and all after it give the ſame value 0.

22. Taking the geometrical ſeries of terms 1 − 2 + 4 − 8 + &c. then [...]

[33]Here the lower parts of all the columns of means, from the cipher 0 downwards, conſiſt of the ſame ſeries of terms + 1 − 1 + 3 − 5 + 11 − 21 + 43 − 85 + &c. and the other part of the columns, from the cipher upwards, as well as each line of oblique means, parallel to, and above the line of ciphers, forms a ſeries of terms ½, ¼, ⅜, 5/16.....⅓ · 2ⁿ ± 1/2ⁿ, alternately above and below the value of the ſeries, ⅓, and approaching continually nearer and nearer to it, and which, when infinitely continued, or when n is infinite, the term becomes ⅓ for the value of the geometrical ſeries, 1 − 2 + 4 − 8 + 16 − &c.

And the ſame ſet of terms would be given by each of the formulae.

23. Take the geometrical ſeries 1 − 3 + 9 − 27 + 81 − &c. Then [...] Here the column of ſucceſſive ſums, and every ſecond column of the arithmetical means, below the o, conſiſts of the ſame ſeries of terms 1, − 2, + 7, − 20, + &c. whilſt all the other columns of means conſiſt of this other ſet of terms ½, − ½, + 2½, − 6½, + &c. alſo the firſt oblique line of means, ½, 0, ½, 0, ½, 0, &c. conſiſts of the terms ½ and 0 alternately, which are all at equal diſtance from the value of the ſeries propoſed 1 − 3 + 9 − 27 + 81 − &c. as indeed are the terms of all the other oblique deſcending lines. And the mean between every two terms gives ¼ for that value. And the ſame terms would be given by the formulae, namely alternately ½ and 0.

And thus the value of any geometrical ſeries, whoſe ratio or ſecond term is r, will be found to be = 1/1+r.

[34]24. Finally, let there be taken the hypergeometrical ſeries 1 − 1 + 2 − 6 + 24 − 120 + &c. = 1 − 1 A + 2 B − 3 C + 4 D − 5 E + &c. which difficult ſeries has been honoured by a very conſiderable memoir written upon the valuation of it by the late famous L. Euler, in the New Peterſburg Commentaries, vol. v. where the value of it is at length determined to be .5963473 &c.

To ſimplify this ſeries, let us omit the firſt two terms 1 − 1 = 0, which will not alter the value, and divide the remaining terms by 2, and the quotients will give 1 − 3 + 12 − 60 + 360 − 2520 + &c. which, being half the propoſed ſeries, ought to have for its value the half of .596347 &c. namely .298174 nearly.

Now, ranging the terms in a column, and taking the ſums and means as uſual, we have [...] Where it is evident, that the diverging is ſomewhat diminiſhed, but not quite counteracted, in the columns and oblique deſcending lines from beginning to end, as the terms in thoſe directions ſtill increaſe, though not quite ſo faſt as the original ſeries; and that the ſigns of the ſame terms are alternately + and −, while thoſe of the terms in the other lines obliquely aſcending from left to right, are alternately one line all +, and another line all −, and theſe terms continually decreaſing. The terms in the oblique deſcending lines, being alternately too great and too little, are the fitteſt to proceed with again. Take therefore any one of thoſe lines, as ſuppoſe the firſt, and ranging it vertically, take the means as before, and they will approach nearer to the value of the ſeries, thus: [35] [...] Here the ſame approximation in the lines and columns, towards the value of the ſeries, is obſervable again, only in a higher degree; alſo the terms in the columns and oblique deſcending lines, are again alter⯑nately too great and too little, but now within narrower limits, and the ſigns of the terms are more of them poſitive; alſo the terms in each oblique aſcending line, are ſtill either all above or all below the value of the ſeries, and that alternately one line after another as before. But the deſcending lines will again be the fitteſt to uſe, becauſe the terms in each are alternately above and below the value ſought. Tak⯑ing therefore again the firſt of theſe oblique deſcending lines, treat it as before, and we ſhall obtain ſets of terms approaching ſtill nearer to the value, thus: [...] Here the approach to an equality, among all the lines and columns, is ſtill more viſible, and the deviations reſtricted within narrower limits, the terms in the oblique aſcending lines ſtill on one ſide of the value, and gradually increaſing, while the columns and the oblique deſcending lines, for the moſt part, have their terms alternately too great and too little, as is evident from their alternately becoming greater and leſs than each other: and from an inſpection of the whole, it is eaſy to pronounce that the firſt three figures of the number ſought, will be 298. Taking therefore the laſt ſour terms of the firſt deſcending line, and proceeding as before, we have [36] [...]

And, finally, taking the loweſt aſcending line, becauſe it has moſt the appearance of being alternately too great and too little, proceed with it as before, thus: [...] where the numbers in the lines and columns gradually approach nearer together, till the laſt mean is true to the neareſt unit in the laſt figure, giving us .298174 for the value of the propoſed hypergeometrical ſeries 1 − 3 + 12 − 60 + 360 − 2520 + 20160 − &c.

And in like manner are we to proceed with any other ſeries whoſe terms have alternate ſigns.

Royal Military Acad. Woolwich, May, 1780.

SINCE the foregoing method was diſcovered, and made known to ſeveral friends, two paſſages have been offered to my conſideration, which I ſhall here mention, in juſtice to their authors, Sir Iſaac New⯑ton, and the late learned Mr. Euler.

The firſt of theſe is in Sir Iſaac's letter to Mr. Oldenburg, dated October 24, 1676, and may be ſeen in Collins's Commercium Epiſtolicum, p. 177, the laſt paragraph near the bottom of the page, namely, Per [37]ſeriem Leibnitii etiam, ſi ultimo loco dimidium termini adjiciatur, & alia quaedam ſimilia artificia adhibeantur, poteſt computum produci ad multas figuras. The ſeries here alluded to, is 1 − ⅓ + ⅕ − 1/7 + 1/9 − 1/11 + &c. denoting the area of the circle whoſe diameter is 1; and Sir Iſaac here directs to add in half the laſt term, after having collected all the foregoing, as the means of obtaining the ſum a little exacter. And this, indeed, is equivalent to taking one arithmetical mean between two ſucceſſive ſums, but it does not reach the idea contained in my method. It appears alſo, both by the other words, & alia quaedam ſimilia artificia adhibeantur, contained in the above extract, and by theſe, alias artes adhibuiſſem, a little higher up in the ſame page 177, that Sir Iſaac Newton had ſeveral other contrivances for obtaining the ſums of ſlowly converging ſeries; but what they were, it may perhaps be now impoſſi⯑ble to determine.

The other is a paſſage in the Novi Comment. Petropol. tom. v. p. 226, where Mr. Euler gives an inſtance of taking one ſet of arithmetical means between a ſeries of quantities which are gradually too little and too great, to obtain a nearer value of the ſum of a ſeries in queſtion. But neither does this reach the idea contained in my method. How⯑ever, I have thought it but juſtice to the characters of theſe two eminent men, to make this mention of their ideas, which have ſome relation to my own, though unknown to me at the time of my diſcovery.

TRACT III.

A Method of ſumming the Series a + bx + cx2 + dx3 + ex4 + &c. when it converges very ſlowly, namely, when x is nearly equal to 1, and the Coefficients a, b, c, d, &c. decreaſe very ſlowly: the Signs of all the Terms being poſitive.

Art. 1. WHEN we have occaſion to find the ſum of ſuch ſeries as that above-mentioned, having the coefficients a, b, c, d, &c. of the terms, decreaſing very ſlowly, and the con⯑verging quantity x pretty large; we can neither find the ſum by col⯑lecting the terms together, on account of the immenſe number of them which it would be neceſſary to collect; neither can it be ſummed by means of the differential ſeries, becauſe the powers of the quantity x/1−x will then diverge faſter than the differential coefficients converge. In ſuch caſe then we muſt have recourſe to ſome other method of tranſ⯑forming it into another ſeries which ſhall converge faſter. The follow⯑ing is a method by which the propoſed ſeries is changed into another, which converges ſo much the quicker as the original ſeries is ſlower.

2. The method is thus. Aſſume a2/D the given ſeries a + bx + cx2 + dx3 + &c. Then ſhall [...]; which, by actual diviſion, is [...] Conſequently a2 divided by this ſeries will be equal to the ſeries propoſed, and this new ſeries will be very eaſily ſummed, in com⯑pariſon with the original one, becauſe all the coefficients after the ſecond term are evidently very ſmall; and indeed they are ſo much the ſmaller, and fitter for ſummation, by how much the coefficients of the [39]original ſeries are nearer to equality; ſo that when theſe a, b, c, d, &c. are quite equal, then the third, fourth, &c. coefficients of the new ſeries become equal to nothing, and the ſum accurately equal to [...]; which we alſo know to be true from other principles.

3. Although the firſt two terms, a − bx, of the new ſeries be very great in compariſon with each of the following terms, yet theſe latter may not always be ſmall enough to be entirely rejected where much accuracy is required in the ſummation. And in ſuch caſe it will be neceſſary to collect a great number of them, to obtain their ſum pretty near the truth; becauſe their rate of converging is but ſmall, it being indeed pretty much like to the rate of the original ſeries, but only the terms, each to each, are much ſmaller, and that commonly in a degree to the hundredth or thouſandth part.

4. The reſemblance of this new ſeries however, beginning with the third term, to the original one, in the law of progreſſion, intimates to us that it will be beſt to ſum it in the very ſame manner as the former. Hence then putting

• a′ = c − b2/a
• b′ = d − 2bc/a + b3/a2
• c′ = e − 2bd+c2/a + 3b2 c/a2 − b4/a3 &c.

and conſequently the propoſed ſeries a + bx + cx2 + &c. = [...], by taking the ſum of the ſeries a′ + b′ x + c′ x2 + &c. by the very [40]ſame theorem as before, the ſum S of the original ſeries will then be expreſſed thus, [...]. where the ſeries in the laſt denominator, having again the ſame pro⯑perties with the former one, will have its firſt two terms very large in reſpect of the following terms. But theſe firſt two terms, a′ − b′x, are themſelves very ſmall in compariſon with the firſt two, a − bx, of the former ſeries; and therefore much more are the third, fourth, &c. terms of this laſt denominator very ſmall in compariſon with the ſame a − bx: for which reaſon they may now perhaps be ſmall enough to be neglected.

5. But if theſe laſt terms be ſtill thought too large to be omitted, then find the ſum of them by the very ſame theorem: and thus proceed, by repeating the operation in the ſame manner, till the required degree of accuracy is obtained. Which it is evident, will happen after a ſmall number of repetitions, becauſe that, in each new denominator, the third, fourth, &c. terms are commonly depreſſed, in the ſcale of num⯑bers, two or three places lower than the firſt and ſecond terms are. And the general theorem, denoting the ſum S when the proceſs is con⯑tinually repeated, will be this, [...].

[41]6. But the general denominator D in the fraction a2/D, which is aſ⯑ſumed for the value of S or a + bx + cx2 + &c. may be otherwiſe found as below; from which the general law of the values of the coefficients will be rendered viſible. Aſſume S or a + bx + cx2 + &c. or [...]; then ſhall [...] Hence, by equating the coefficients of the like terms to nothing, we obtain the following general values:

• a′ = c − bb/a
• [...]
• [...]
• [...]
• [...] &c.

Where the values of the coefficients are the ſame in effect as before found, but here the law of their continuation is manifeſt

7. To exemplify now the uſe of this method, let it be propoſed to ſum the very ſlow ſeries x + ½x2 + ⅓x3 + ¼x4 + ⅕x5 + ⅙x6 + &c. when x = 9/10 = .9, which denotes the hyperbolic log. of 1/1−x, or in this caſe of 10.

[42]Now it will be proper, in the firſt place, to collect a few of the firſt terms together, and then apply the theorem to the remaining terms, which, by being nearer to an equality than the terms are near the be⯑ginning of the ſeries, will be fitter to receive the application of the theorem. Thus to collect the firſt 12 terms:

Then we have to find the ſum of the reſt of the terms after theſe firſt 12, namely of x13 × : 1/13 + 1/14x + 1/15x2 + 1/16x3 + &c. in which x = .9, and x13 = .2541865828329; alſo a = 1/13, b = 1/14, c = 1/15, &c. and the firſt application of our rule, gives, for the value of 1/13 + 1/14x + 1/15x2 + &c. or S, [...] the ſecond gives [...] the third gives [...] [43]the fourth gives [...] Or, when the terms in the numerators are ſquared, it is [...] Or, by omitting a proper number of ciphers, it is [...]

I have written an unknown quantity z after the laſt denominator, to repreſent the ſmall quantity to be ſubtracted from the laſt denominator 344. Now, rejecting the ſmall quantity z, and beginning at the laſt fraction to calculate, their values will be as here ranged in the firſt an⯑nexed column.

[...] [44]placing z below them for the next unknown fraction. Divide then every fraction by the next below it, placing the quotients or ratios in the next column. Then take the quotients or ratios of theſe; and ſo on till the laſt ratio [...]; which, from the nature of the ſeries of the firſt terms of every column, muſt be leſs than the next preceding one 2.39: conſequently z muſt be leſs than 1.68×187/63, or leſs than 5. But, from the nature of the ſeries in the vertical row or column of firſt ratios, 187/z muſt be leſs than 63; and conſequently z muſt be greater than 187/63, or greater than 3. Since then z is leſs than 5 and greater than 3, it is probable that the mean value 4 is near the truth: and accordingly taking 4 for z, or rather 4.3, as z appears to be nearer 5 than 3, and taking the continual ratios, as placed along the laſt line of the table, their values are found to accord very well with the next preceding numbers, both in the columns and oblique rows.

Hence, uſing 043 for z in the denominator .344 − z of the laſt frac⯑tion of the general expreſſion, and computing from the bottom, upwards through the whole, the quotients, or values of the fractions, in the inverted order, will be

• 213
• 12079
• 1223397
• .518414000

of which the laſt muſt be nearly the value of the ſeries 1/13 + 1/14x + 1/15x2 + &c. when x = .9.

Then this value .518414 of the ſeries, being multiplied by x13 or .2541865828329, gives .1317738 for the ſum of all the terms of the original ſeries after the firſt 12 terms, to which therefore the ſum of the firſt 12 terms, or 2.17081162, being added, we have 2.30258542 for the ſum of the original ſeries x + ½x2 + ⅓x3 + ¼x4 + &c. Which value is true within about 3 in the 8th place of figures, the more accurate value being 2.30258509 &c. or the hyp. log. of 10.

TRACT IV. The Inveſtigation of certain eaſy and General Rules, for Extracting any Root of a given Number.

1. THE roots of given numbers are commonly to be found, with much eaſe and expedition, by means of logarithms, when the indices of ſuch roots are ſimple numbers, and the roots are not required to a great number of figures. And the ſquare or cubic roots of numbers, to a good practical degree of accuracy, may be ob⯑tained, almoſt by inſpection, by means of my tables of ſquares and cubes, publiſhed by order of the Commiſſioners of Longitude, in the year 1781. But when the indices of ſuch roots are certain complex or irrational numbers, or when the roots are required to be found to a great many places of figures, it is neceſſary to make uſe of certain ap⯑proximating rules, by means of the ordinary arithmetical computa⯑tions. Such rules as are here alluded to, have only been diſcovered ſince the great improvements in the modern algebra: and the perſons who have beſt ſucceeded in their enquiries after ſuch rules, have been ſucceſſively Sir Iſaac Newton, Mr. Raphſon, M. de Lagney, and Dr. Halley; who have ſhewn that the inveſtigation of ſuch theorems is alſo uſeful in diſcovering rules for approximating to the roots of all ſorts of affected algebraical equations, to which the former rules, for the roots of all ſimple equations, bear a conſiderable affinity. It is preſumed that the following ſhort tract contains ſome advantages over any other method that has hitherto been given, both as to the eaſe and univerſality of the concluſions, and the general way in which the inveſtigations are made: for here, a theorem is diſcovered, which, although it be general for all roots whatever, is at the ſame time very accurate, and ſo ſimple and [46]eaſy to uſe and to keep in mind, that nothing more ſo can be deſired or hoped for; and farther, that inſtead of ſearching out rules ſeverally for each root, one after another, our inveſtigation is at once for any indefinite poſſible root, by whatever quantity the index is expreſſed, whether fractional, or irrational, or ſimple, or compound.

2. In every theorem, or rule, here inveſtigated, let

• N be the given number, whoſe root is ſought,
• n the index of that root,
• a its neareſt rational root, or aⁿ; the neareſt rational power to N, whe⯑ther greater or leſs,
• x the remaining part of the root ſought, which may be either poſitive or negative, namely, poſitive when N is greater than aⁿ, otherwiſe negative.

Hence then the given number N is [...], and the required root [...] = a + x.

3. Now, for the firſt rule, expand the quantity [...] by the binomial theorem, ſo ſhall we have [...] Subtract aⁿ from both ſides, ſo ſhall [...] Divide by [...], ſo ſhall [...] or [...] Here, on account of the ſmallneſs of the quantity x in reſpect of a, all the terms of this ſeries, after the firſt term, will be very ſmall, and may therefore be neglected without much error, which gives us [...] for a near value of x, being only a ſmall matter too great. And conſequently [...] is nearly = N¹/n the root ſought. And this may be accounted the firſt theorem.

[47]4. Again, let the equation [...] be multi⯑tiplied by n − 1, and aⁿ added to each ſide, ſo ſhall we have [...] for a diviſor: Alſo multiply the ſides of the ſame equation by a and ſubtract aⁿ + 1 from each, ſo ſhall we have [...] for a dividend: Divide now this dividend by the diviſor, ſo ſhall [...] Which will be nearly equal to x, for the ſame reaſon as before; and this ex⯑preſſion is nearly as much too little as the former expreſſion was too great. Conſequently, by adding a, we have a + x or N¹/n nearly [...] for a ſecond theorem, and which is nearly as much in defect as the for⯑mer was in exceſs.

5. Now becauſe the two foregoing theorems differ from the truth by nearly equal ſmall quantities, if we add together the two numerators and the two denominators of the foregoing two fractional expreſſions, namely [...] and [...], the ſums will be the numerator and denominator of a new fraction, which will be much nearer than either of the former. The fraction ſo found is [...]; which will be very nearly equal to N¹/n or a + x the root ſought; for, by diviſion, it is found to be equal to a + x * − n−1/2 · n+1/6 · x3/a2 + &c. where the term is wanting which contains the ſquare of x, and the following terms are very ſmall. And this is the third theorem.

6. A fourth theorem might be found by taking the arithmetical mean [48]between the firſt and ſecond, which would be [...]; which will be nearly of the ſame value, though not ſo ſimple, as the third theorem; for this arithmetical mean is found equal to a + x * + n−1/2 · n−2/3 · x3/a2 + &c.

7. But the third theorem may be inveſtigated in a more general way, thus: Aſſume a quantity of this form [...], with coefficients p and q to be determined from the proceſs; the other letters N, a, n, repre⯑ſenting the ſame things as before; then divide the numerator by the denominator, and make the quotient equal to a + x, ſo ſhall the compariſon of the coefficients determine the relation between p and q required. Thus, [...] [...] then dividing the former of theſe by the latter, we have [...] or [...] Then, by equating the correſponding terms, we obtain theſe three equations

• [...] = a,
• p−q/p+q n = 1,
• n−1/2 − qn/p+q = 0.

From which we find p−q/p+q = 1/n and p ∶ q ∷ n + 1 ∶ n − 1. So that by ſubſtituting n + 1 and n − 1, or any quantities proportional to them, for p and q, we ſhall have [...] for the va⯑lue of the aſſumed quantity [...], which is ſuppoſed nearly equal to a + x, the required root of the quantity N.

[49]8. Now this third theorem [...], which is ge⯑neral for roots, whatever be the value of n, and whether aⁿ be greater or leſs than N, includes all the rational formulas of De Lagney and Halley, which were ſeparately inveſtigated by them; and yet this ge⯑neral formula is perfectly ſimple and eaſy to apply, and eaſier kept in mind than any one of the ſaid particular formulas. For, in words at length, it is ſimply this: to n + 1 times N add n − 1 times aⁿ, and to n − 1 times N add n + 1 times aⁿ, then the former ſum multi⯑plied by a and divided by the latter ſum, will give the root N¹/n nearly; or, as the latter ſum is to the former ſum, ſo is a, the aſſumed root, to the required root, nearly. Where it is to be obſerved that aⁿ may be taken either greater or leſs than N, and that the nearer it is to it, the better.

9. By ſubſtituting for n, in the general theorem, ſeverally the num⯑bers 2, 3, 4, 5, &c. we ſhall obtain the following particular theorems, as adapted for the 2d, 3d, 4th, 5th, &c. roots, namely, for the

• 2d or ſquare root, [...]
• 3d or cube root, [...]
• 4th root [...]
• 5th root [...]
• 6th root [...]
• 7th root [...]

10. To exemplify now our formula, let it be firſt required to extract the ſquare root of 365. Here N = 365, n = 2; the neareſt ſquare is 361, whoſe root is 19.

[50]Hence 3 N + a2 = 3 × 365 + 361 = 1456, and N + 3 a2 = 365 + 3 × 361 = 1448; then as 1448 ∶ 1456 ∷ 19 ∶ 19×182/181 = 19 19/181 = 19.10497 &c.

Again, to approach ſtill nearer, ſubſtitute this laſt found root 19×182/181 for a, the values of the other letters remaining as before, we have a2 = 19²×182²/181² = 3458²/181²; then

• 3N + a2 = 3 × 365 + 3458²/181² = 47831059/32761,
• N + 3a2 = 365 + 3×3458²/181² = 47831057/32761;

hence 47831057 ∶ 47831059 ∷ 19×182/181 or 3458/181 ∶ 3458×47831059/181×47831057 = the root of 365 very exact, which being brought into decimals, would be true to about 20 places of figures.

11. For a ſecond example, let it be propoſed to double the cube, or to find the cube root of the number 2.

Here N = 2, n = 3, the neareſt root a = 1, alſo a3 = 1. Hence 2 N + a3 = 4 + 1 = 5, and N + 2 a3 = 2 + 2 = 4; then as 4 ∶ 5 ∷ 1 ∶ 5/4 = 1.25 = the firſt approximation. Again, take a = 5/4, and conſequently a3 = 125/64; Hence 2N + a3 = 4 + 125/64 = 381/64, and N + 2a3 = 2 + 250/64 = 378/64; then as 378 : 381, or as 126 ∶ 127 ∷ 5/4 ∶ 5/4 × 127/126 = 635/504 = 1.259921, for the cube root of 2, which is true in the laſt figure.

And by taking 635/504 for the value of a, and repeating the proceſs, a great many more figures may be found.

[51]12. For a third example, let it be required to find the 5th root of 2.

Here N = 2, n = 5, the neareſt root a = 1.

Hence 3 N + 2 a5 = 6 + 2 = 8, and 2 N + 3 a5 = 4 + 3 = 7; then as 7 ∶ 8 ∷ 1 ∶ 8/7 = 1 1/7 for the firſt approximation.

Again, taking a = 8/7, we have 3 N + 2 a5 = 6 + 65536/16807 = 166378/16807, 2 N + 3 a5 = 4 + 98304/16807 = 165532/16807; then as 165532 ∶ 166378 ∷ 8/7 ∶ 8/7 × 83189/82766 = 4/7 × 83189/41383 = 332756/289681 = 1.148698 &c. for the 5th root of 2, and is true in the laſt figure.

13. To find the 7th root of 126⅓.

Here N = 126⅕, n = 7, the neareſt root a = 2, alſo a7 = 128.

Hence 4 N + 3 a7 = 504⅘ + 384 = 888⅘ = 4444/5, and 3 N + 4 a7 = 378⅗ + 512 = 890⅗ = 4453/5; then as 4453 ∶ 4444 ∷ 2 ∶ 8888/4453 = 1.995957, for the root very exact by one operation, being true to the neareſt unit in the laſt figure.

14. To find the 365th root of 1.05, or the amount of 1 pound for 1 day, at 5 per cent. per annum, compound intereſt.

Here N = 1.05, n = 365, a = 1 the neareſt root. Hence 366 N + 364 a = 748.3, and 364 N + 366 a = 748.2; then as 748.2 ∶ 784.3 ∷ 1 ∶ 7483/7482 = 1 1/7482 = 1.00013366, the root ſought very exact at one operation.

15. Let it be required to find the value of the quantity [...] or [...].

[52]Now this may be done two ways; either by finding the ⅔ power or 3/2 root of 21/4 at once; or elſe by finding the 3d or cubic root of 21/4, and then ſquaring the reſult.

By the firſt way:—Here it is eaſy to ſee that a is nearly = 3, becauſe 3³/2 = √27 = 5 + ſome ſmall fraction. Hence, to find nearly the ſquare root of 27, or √27, the neareſt power to which is 25 = a2 in this caſe: Hence 3 N + a2 = 3 × 27 + 25 = 106, and N + 3 a2 = 27 + 3 × 25 = 102; then as 102 : 106, or as 51 ∶ 53 ∷ 5 ∶ 5 × 53/51 = 265/51 = √ 27 nearly.

Then having N = 21/4, n = 3/2, a = 3, and a3/2 = 265/51 nearly; it will be 5/2 N + ½ a3/2 = 5/2 × 21/4 + ½ × 265/51 = 6415/408, and ½ N + 5/2 a5/2 = ½ × 21/4 + 5/2 × 265/51 = 6371/408; hence as 6371 ∶ 6415 ∷ 3 ∶ 19245/6371 = 3 134/6371 = 3.020719, for the value of the quantity ſought nearly, by this way.

Again, by the other method, in finding firſt the value of [...], or the cube root of 21/4. It is evident that 2 is the neareſt integer root, being the cube root of 8 = a3.

Hence 2 N + a3 = 21/2 + 8 = 74/4, and N + 2 a3 = 21/4 + 16 = 85/4; then as 85 ∶ 74 ∷ 2 ∶ 148/85 or = 7/4 nearly. Then taking 7/4 for a, we have 2 N + a3 = 21/2 + 343/64 = 1015,64, and N + 2 a3 = 21/4 + 2.343/64 = 1022/64; [53]hence as 1022 : 1015, or as [...] nearly. Conſequently the ſquare of this, or [...] will be = 7²/4² × 145²/146² = 1030225/341056 = 3 7057/341056 = 3.020690, the quantity ſought more nearly, being true in the laſt figure.

TRACT V. A new Method of finding, in finite and general Terms, near Values of the Roots of Equations of this Form, [...]; namely, having the Terms alternately Plus and Minus.

1. THE following is one method more, to be added to the many we are already poſſeſſed of, for determining the roots of the higher equations. By means of it we readily find a root, which is ſometimes accurate; and when not ſo, it is at leaſt near the truth, and that by an eaſy finite formula, which is general for all equations of the above form, and of the ſame dimenſion, provided that root be a real one. This is of uſe for depreſſing the equation down to lower dimen⯑ſions, and thence for finding all the roots one after another, when the formula gives the root ſufficiently exact; and when not, it ſerves as a ready means of obtaining a near value of a root, by which to com⯑mence an approximation ſtill nearer, by the previouſly known methods of Newton, or Halley, or others. This method is farther uſeful in elucidating the nature of equations, and certain properties of numbers; as will appear in ſome of the following articles. We have already eaſy methods for finding the roots of ſimple and quadratic equations. I ſhall therefore begin with the cubic equation, and treat of each order of equations ſeparately, in aſcending gradually to the higher dimenſions.

2. Let then the cubic equation x3 − px2 + qx − r = o be pro⯑poſed. Aſſume the root x = a, either accurately or approximately, as it may happen, ſo that x − a = o, accurately or nearly. Raiſe this [55] x − a = o to the third power, the ſame dimenſion with the propoſed equation, ſo ſhall x3 − 3 a x2 + 3 a2 x − a3 = o; but the propoſed equation is x3 − p x2 + q x − r = o; therefore the one of theſe is equal to the other. But the firſt term (x3) of each is the ſame; and hence, if we aſſume the ſecond terms equal between themſelves, it will follow that the ſum of the two remain⯑ing terms will alſo be equal, and give a ſimple equation by which the value of x is determined. Thus, 3a x2 being = px2, or a = ⅓p, we ſhall have 3a2 x − a3 = qx − r, and hence [...], by ſubſtituting ⅓p, the value of a, inſtead of it.

3. Now this value of x here found, will be the middle root of the propoſed cubic equation. For becauſe a is aſſumed nearly or accu⯑rately equal to x, and alſo equal to ⅓ p, therefore x is = ⅓ p nearly or accurately, that is, ⅓ of the ſum of the three roots, to which the coefficient p of the ſecond term of the equation, is always equal; and thus, being a medium among the three roots, it will be either nearly or accurately equal to the middle root of the propoſed equation, when that root is a real one.

4. Now this value of x will always be the middle root accurately, whenever the three roots are in arithmetical progreſſion; otherwiſe, only approximately. For when the three roots are in arithmetical progreſ⯑ſion, ⅓ p or ⅓ of their ſum, it is well known, is equal to the middle term or root. In the other caſes, therefore, the above-found value of x is only near the middle root.

5. When the roots are in arithmetical progreſſion, becauſe the middle term or root is then = ⅓p, and alſo [...], therefore [...], or [...], an equa⯑tion [56]expreſſing the general relation of p, q, and r; where p is the ſum of any three terms in arithmetical progreſſion, q the ſum of their three rectangles, and r the product of all the three. For, in any equation, the coefficient p of the ſecond term, is the ſum of the roots; the coefficient q of the third term, is the ſum of the rectangles of the roots; and the coefficient r of the fourth term, is the ſum of the ſolids of the roots, which in the caſe of the cubic equation is only one:—Thus, if the roots, or arithmetical terms, be 1, 2, 3. Here p = 1 + 2 + 3 = 6, q = 1 × 2 + 1 × 3 + 2 × 3 = 2 + 3 + 6 = 11, r = 1 × 2 × 3 = 6; then 2 p3 = 2 × 6³ = 432, and [...] alſo.

6. To illuſtrate now the rule [...] by ſome examples; let us in the firſt place take the equation x3 − 6 x2 + 11 x − 6 = 0. Here p = 6, q = 11, and r = 6; conſequently [...]. This being ſubſtituted for x in the given equation, makes all the terms to vaniſh, and therefore it is an exact root, and the roots will be in arithmetical progreſſion. Dividing therefore the given equation by x − 2 = 0, the quotient is x2 − 4x + 3 = 0, the roots of which quadratic equation are 3 and 1, the other two roots of the propoſed equation x3 − 6 x2 + 11 x − 6 = 0.

7. If the equation be x3 − 39x2 + 479x − 1881 = 0; we ſhall have p = 39, q = 479, and r = 1881; then [...]. Then, ſubſtituting 11 2/7 for x in the propoſed equation, the negative terms are ſound to exceed the poſitive terms by 5, thereby ſhewing that 11 2/7 is very near, but ſomething above, the middle root, and that there⯑fore the roots are not in arithmetical progreſſion. It is therefore pro⯑bable [57]that 11 may be the true value of the root, and on trial it is found to ſucceed.

Then dividing x3 − 39x2 + 479x − 1881 by x − 11, the quotient is x• − 28x + 171 = 0, the roots of which quadratic equation are 9 and 19, the two other roots of the propoſed equation.

8. If the equation be x2 − 6x2 + 9x − 2 = 0; we ſhall have p = 6, q = 9, and r = 2; then [...]. This value of x being ſubſtituted for it in the propoſed equation, cauſes all the terms to vaniſh, as it ought, thereby ſhewing that 2 is the middle root, and that the roots are in arithmetical progreſſion.

Accordingly, dividing the given quantity x3 − 6x2 + 9x − 2 by x − 2, the quotient is x• − 4x + 1 = 0, a quadratic equation, whoſe roots are 2 + √2 and 2 − √2, the two other roots of the equation propoſed.

9. If the equation be x3 − 5x2 + 5x − 1 = 0; we ſhall have p = 5, q = 5, and r = 1; then [...]. From which one might gueſs the root ought to be 1, and which being tried, is found to ſucceed.

But without ſuch trial, we may make uſe of this value 1 4/45, or 1 1/ [...] nearly, and approximate with it in the common way.

Having found the middle root to be 1, divide the given quantity x3 − 5x2 + 5x − 1 by x − 1, and the quotient is x2 − 4x + 1 = 0, the roots of which are 2 + √2 and 2 − √2, the two other roots, as in the laſt article.

[58]10. If the equation be x3 − 7x2 + 18x − 18 = 0; we ſhall have p = 7, q = 18, and r = 18; then [...] or 3 nearly. Then trying 3 for x, it is found to ſucceed. And dividing x3 − 7x2 + 18x − 18 by x − 3, the quotient is x• − 4x + 6 = 0, a quadratic equation whoſe roots are 2 + √−2 and 2 − √−2, the two other roots of the propoſed equation, which are both impoſſible or imaginary.

11. If the equation be x3 − 6x2 + 14x − 12 = 0; we ſhall have p = 6, q = 14, and r = 12; then [...]. Which being ſubſti⯑tuted for x, it is found to anſwer, the ſum of the terms coming out = 0. Therefore the roots are in arithmetical progreſſion. And, ac⯑cordingly, by dividing x3 − 6x2 + 14x − 12 by x − 2, the quotient is x2 − 4x + 6 = 0, the roots of which quadratic equation are 2 + √−2 and 2 − √−2, the two other roots of the propoſed equation, and the common difference of the three roots is √−2.

12. But if the equation be x3 − 8x2 + 22x − 24 = 0; we ſhall have p = 8, q = 22, and r = 24; then [...]. Which being ſubſtituted for x in the propoſed equation, the ſum of the terms differs very widely from the truth, thereby ſhewing that the middle root of the equation is an imaginary one, as it is indeed, the three roots being 4, and 2 + √−2, and 2 − √−2.

13. In Art. 2 the value of x was determined by aſſuming the ſecond terms of the two equations equal to each other. But a like near value might be determined by aſſuming either the two third terms, or the two ſourth terms equal.

[59]Thus the equations being

• x3 − 3ax2 + 3a2 x − a3 = 0,
• x3 − px2 + qx − r = 0,

if we aſſume the third terms 3a2 x and qx equal, or a = √⅓q, the ſums of the ſecond and fourth terms will be equal, namely, 3ax2 + a3 = px2 + r; and hence we find [...] by ſubſtituting √⅓q the value of a inſtead of it.

And if we aſſume the fourth terms equal, namely a3 = r, or 3√r, then the ſums of the ſecond and third terms will be equal, namely, 3ax − 3a2 = px − q; and hence [...], by ſubſtitu⯑ting r⅓ inſtead of a. And either of theſe two formulas will give nearly the ſame value of the root as the firſt formula, at leaſt when the roots do not differ very greatly from one another.

But if they differ very much among themſelves, the firſt formula will not be ſo accurate as theſe two others, becauſe that in them the roots were more complexly mixed together; for the ſecond formula is drawn from the coefficient of the third term, which is the ſum of all the rec⯑tangles of the roots; and the third formula is drawn from the coefficient of the laſt term, which is equal to the continual product of all the roots; while the firſt formula is drawn from the coefficient of the ſecond term, which is ſimply the ſum of the roots. And indeed the laſt theorem is commonly the neareſt of all. So that when we ſuſpect the roots to be very wide of each other, let either the ſecond or third be uſed.

Thus, in the equation x3 − 23x2 + 62x − 40 = 0, whoſe three roots are 1, 2, and 20. Here p = 23, q = 62, r = 40; and by

• the 1ſt theor. [...] nearly,
• 2d theor. [...] nearly,
• 3d theor. [...] nearly.

Where the two latter are much nearer the middle root (2) than the firſt. [60]And the mean between theſe two is 2 1/42, which is very near to that root. And this is commonly the caſe, the one being nearly as much too great as the other is too little.

14. To proceed now, in like manner, to the biquadratic equation, which is of this general form x4 − px3 + qx2 − rx + s = 0.

Aſſume the root x = a, or x − a = 0, and raiſe this equation x − a = 0 to the fourth power, or the ſame height with the propoſed equation, which will give x4 − 4ax3 + 6a2 x2 − 4a3 x + a4 = 0; but the propoſed equation is x4 − px3 + qx2 − rx + s = 0; therefore theſe two are equal to each other. Now if we aſſume the ſecond terms equal, namely 4a = p, or a = ¼p, then the ſums of the three remaining terms will alſo be equal, namely, [...]; and hence [...], or [...] by ſubſtituting ¼p inſtead of a: then, reſolving this quadratic equation, we find its roots to be thus [...]; or if we put A = 3/2 p2 − 4q, B = p2 − 16r, C = p4 − 256s, the two roots will be [...].

15. It is evident that the ſame property is to be underſtood here, as for the cubic equation in Art. 3, namely, that the two roots above found, are the middle roots of the four which belong to the biquadratic equation, when thoſe roots are real ones; for otherwiſe the formulae are [61]of no uſe. But however thoſe roots will not be accurate, when the ſum of the two middle roots, of the propoſed equation, is equal to the ſum of the greateſt and leaſt roots, or when the four roots are in arithmetical progreſſion; becauſe that, in this caſe, ¼ p, the aſſumed value of a, is neither of the middle roots exactly, but only a mean between them.

16. To exemplify this formula [...], let the propo⯑ſed equation be x4 − 12 x3 + 49 x2 − 78 x + 40 = 0. Then A = 3/2 p2 − 4 q = 12² × 3/2 − 4 × 49 = 216 − 196 = 20, B = p3 − 16 r = 12³ − 16 × 78 = 1728 − 1248 = 480, C = p4 − 256s = 12⁴ − 256 × 40 = 20736 − 10240 = 10496. Hence [...] nearly, or 4¼ and 1¾ nearly, or nearly 4 and 2, whoſe ſum is 6. And trying 4 and 2, they are both found to anſwer, and there⯑fore they are the two middle roots.

Then [...], by which dividing the given equation x4 − 12 x3 + 49 x2 − 78 x + 40 = 0, the quotient is x2 − 6 x + 5 = 0, the roots of which quadratic equation are 5 and 1, and which therefore are the greateſt and leaſt roots of the equation propoſed.

17. If the equation be x4 − 12 x3 + 47 x2 − 72 x + 36 = 0; then A = 3/2 p2 − 4 q = 12² × 3/2 − 4 × 47 = 216 − 188 = 28, B = p3 − 16 r = 12³ − 16 × 72 = 1728 − 1152 = 576, C = p4 − 256 s = 12⁴ − 256 × 36 = 20736 − 9216 = 11520. Hence [...] and 2 1/7, or 3 and 2 nearly; both of which anſwer on trial; and therefore 3 and 2 are the two middle roots.

[62]Then [...], by which dividing the given quantity x4 − 12 x3 + 47 x2 − 72 x + 36 = 0, the quotient is x2 − 7 x + 6 = 0, the roots of which quadratic equation are 6 and 1, which therefore are the greateſt and leaſt roots of the equation propoſed.

18. If the equation be x4 − 7 x3 + 15 x2 − 11 x + 3 = 0; we have A = 3/2 p2 − 4 q = 7² × 3/2 − 4 × 15 = 73½ − 60 = 13½, B = p3 − 16 r = 7³ − 16 × 11 = 343 − 176 = 167, C = p4 − 256 s = 7⁴ − 256 × 3 = 2401 − 768 = 1633. Hence [...] or nearly 2 and 1; both which are found, on trial, to anſwer; and therefore 2 and 1 are the two middle roots ſought.

Then [...], by which dividing the given equation x4 − 7 x3 + 15 x2 − 11 x + 3 = 0, the quotient is x2 − 4 x + 1 = 0, the roots of which quadratic equation are 2 + √2 and 2 − √2, and which therefore are the greateſt and leaſt roots of the propoſed equation.

19. But if the equation be x4 − 9 x3 + 30 x2 − 46 x + 24 = 0; we have

• A = 3/2p2 − 4 q = 9² × 3/2 −4 × 30 = 121½ − 120 = 1½,
• B = p3 − 16 r = 9³ − 16 × 46 = 729 − 736 = − 7,
• C = p4 − 256 s = 9⁴ − 256 × 24 = 6561 − 6144 = 417.

Hence [...], an imaginary quantity, ſhewing that the two middle roots are imaginary, and therefore the formula is of no uſe in this caſe, the four roots being 1, 2 + √ −2, 2 − √ −2, and 4.

20. And thus in other examples the two middle roots will be found when they are rational, or a near value when irrational, which in this [63]caſe will ſerve for the foundation of a nearer approximation, to be made in the uſual way.

We might alſo find another formula for the biquadratic equation, by aſſuming the laſt terms as equal to each other; for then the ſum of the 2d, 3d, and 4th terms of each would be equal, and would form another quadratic equation, whoſe roots would be nearly the two mid⯑dle roots of the biquadratic propoſed.

21. Or a root of the biquadratic equation may eaſily be found, by aſſuming it equal to the product of two ſquares, as [...]. For, comparing the terms of this with the terms of the equation pro⯑poſed, in this manner, namely, making the ſecond terms equal, then the third terms equal, and laſtly the ſums of the fourth and fifth terms equal, theſe equations will determine a near value of x by a ſimple equation. For thoſe equations are [...], [...], [...]. Then the values of ab and a + b, found from the firſt and ſecond of theſe equations, and ſubſtituted in the third, this gives [...], a general formula for one of the roots of the biquadratic equation x4 − px3 + qx2 − rx + s = 0.

22. To exemplify now this ſormula, let us take the ſame equation as in Art. 17, namely, x4 − 12 x3 + 47 x2 − 72 x + 36 = 0, the roots of which were there found to be 1, 2, 3, and 6. Then, by our laſt for⯑mula we ſhall have [...], or nearly 1, which is the leaſt root.

[64]23. Again, in the equation x4 − 7 x3 + 15 x − 11 x2 + 3 = 0, whoſe roots are 1, 2, 2 + √2, and 2 − √2, we have [...] nearly, which is nearly a mean between the two leaſt roots 1 and 2 − √2 or ⅗ nearly.

24. But if the equation be x4 − 9 x3 + 30 x2 − 46 x + 24 = 0, which has impoſſible roots, the four roots being 1, 2 + √−2, 2 − √−2, and 4; we ſhall have [...] nearly, which is of no uſe in this caſe of imaginary roots.

25. This formula will alſo ſometimes fail when the roots are all real. As if the equation be x4 − 12 x3 + 49 x2 − 78 x + 40 = 0, the roots of which are 1, 2, 4, and 5. For here [...], which is of no uſe.

26. For equations of higher dimenſions, as the 5th, the 6th, the 7th, &c. we might, in imitation of this laſt method, combine other forms of quantities together. Thus, for the 5th power, we might compare it either with [...], or with [...], or with [...], or with [...]. And ſo for the other powers.

TRACT VI. Of the Binomial Theorem. With a Demonſtration of the Truth of it in the General Caſe of Fractional Exponents.

1. IT is well known that this famous theorem is called binomial, becauſe it contains a propoſition of a quantity conſiſting of two terms, as a radix, to be expanded in a ſeries of equal value. It is alſo called emphatically the Newtonian theorem, or Newton's binomial theorem, becauſe he has commonly been reputed the author of it, as he was indeed for the caſe of fractional exponents, which is the moſt general of all, and includes all the other particular caſes, of powers, or diviſions, &c.

2. The binomial, as propoſed in its general form, was, by Newton, thus expreſſed [...]; where P is the firſt term of the binomial, Q the quotient of the ſecond term divided by the firſt, and conſe⯑quently PQ is the ſecond term itſelf; or PQ may repreſent all the terms of a multinomial, after the firſt term, and conſequently Q the quotient of all thoſe terms, except the firſt term, divided by that firſt term, and may be either poſitive or negative; alſo m/n repreſents the exponent of the binomial, and may denote any quantity, integral or fractional, poſitive or negative, rational or ſurd. When the exponent [66]is integral, the denominator n is equal to 1, and the quantity then in this form [...], denotes a binomial to be raiſed to ſome power; the ſeries for which was fully determined before Newton's time, as I have ſhewn in the hiſtorical introduction to my Mathematical Tables, lately publiſhed. When the exponent is fractional, m and n may be any quantities whatever, m denoting the index of ſome power to which the binomial is to be raiſed, and n the index of the root to be extracted of that power: and to this caſe it was firſt extended and applied by Newton. When the exponent is negative, the reciprocal of the ſame quantity is meant; as [...] is equal to [...].

3. Now when the radical binomial is expanded in an equivalent ſeries, it is aſſerted that it will be in this general form, namely [...]. where the law of the progreſſion is viſible, and the quantities P, m, n, Q, include their ſigns + or −, the terms of the ſeries being all poſitive when Q is poſitive, and alternately poſitive and negative when Q is negative, independent however of the effect of the coefficients made up of m and n: alſo A, B, C, D, &c. in the latter form, denote each pre⯑ceding term. This latter form is the eaſier in practice, when we want [67]to collect the ſum of the terms of a ſeries; but the former is the fitter for ſhewing the law of the progreſſion of the terms.

4. The truth of this ſeries was not demonſtrated by Newton, but only inferred by way of induction. Since his time however, ſeveral attempts have been made to demonſtrate it, with various ſucceſs, and in various ways; of which however thoſe are juſtly preferred, which proceed by pure algebra, and without the help of fluxions. And ſuch has been eſteemed the difficulty of proving the general caſe independent of the doctrine of fluxions, that many eminent mathematicians to this day account the demonſtration not fully accompliſhed, and ſtill a thing greatly to be deſired. Such a demonſtration I think I have effected. But before I deliver it, it may not be improper to premiſe ſomewhat of the hiſtory of this theorem, its riſe, progreſs, extenſion, and demon⯑ſtrations.

5. Till very lately the prevailing opinion has been, that the theorem was not only invented by Newton, but firſt of all by him; that is, in that ſtate of perfection in which the terms of the ſeries for any aſſigned power whatever, can be found independently of the terms of the pre⯑ceding powers; namely, the ſecond term from the firſt, the third term from the ſecond, the fourth term from the third, and ſo on, by a gene⯑ral rule. Upon this point I have already given an opinion in the hiſtory to my logarithms, above cited, and I ſhall here enlarge ſome⯑what farther on the ſame head.

That Newton invented it himſelf, I make no doubt. But that he was not the firſt inventor, is at leaſt as certain. It was deſcribed by Briggs, in his Trigonometria Britannica, long before Newton was born; not indeed for fractional exponents, for that was the application of Newton, but for any integral power whatever, and that by the general law of the terms as laid down by Newton, independent of the terms of the powers preceding that which is required. For as to the generation of the coefficients of the terms of one power from thoſe of [68]the preceding powers, ſucceſſively one after another, it was remarked by Vieta, Oughtred, and many others, and was not unknown to much more early writers on arithmetic and algebra, as will be manifeſt by a ſlight inſpection of their works, as well as the gradual advance the pro⯑perty made, both in extent and perſpicuity, under the hands of the ſucceſſive maſters in arithmetic, every one adding ſomewhat more towards the perfection of it.

6. Now the knowledge of this property of the coefficients of the terms in the powers of a binomial, is at leaſt as old as the practice of the ex⯑traction of roots; for this property was both the foundation, the prin⯑ciple, and the means of thoſe extractions. And as the writers on arith⯑metic became acquainted with the nature of the coefficients in powers ſtill higher, juſt ſo much higher did they extend the extraction of roots, ſtill making uſe of this property. At firſt it ſeems they were only ac⯑quainted with the nature of the ſquare, which conſiſts of theſe three terms, 1, 2, 1; and accordingly extracted the ſquare roots of numbers by means of them; but went no farther. The nature of the cube next preſented itſelf, which conſiſts of theſe four terms, 1, 3, 3, 1; and by means of theſe they extracted the cubic roots of numbers, in the ſame manner as we do at preſent. And this was the extent of their extrac⯑tions in the time of Lucas de Burgo, an Italian, who, from 1470 to 1500, wrote ſeveral tracts on arithmetic, containing the ſum of what was then known of this ſcience, which chiefly conſiſted in the doctrine of the proportions of numbers, the nature of figurate numbers, and the extraction of roots, as far as the cubic root incluſively.

7. It was not long however before the nature of the coefficients of all the higher powers became known, and tables formed for conſtructing them indefinitely. For in the year 1543 came out, at Norimberg, an excellent treatiſe of arithmetic and algebra, by Michael Stifelius, a Ger⯑man divine, and an honeſt, but a weak, diſciple of Luther. In this work, Arithmetica Integra, of Stifelius, are contained ſeveral curious [69]things, ſome of which have been aſcribed to a much later date. He here treats pretty fully and ably, of progreſſional and figurate numbers, and in particular of the following table for conſtructing both them and the coefficients of the terms of all powers of a binomial, which has been ſo often uſed ſince his time for theſe and other purpoſes, and which more than a century after was, by Paſcal, otherwiſe called the arithme⯑tical triangle, and who only mentioned ſome additional properties of the table.

Stifelius here obſerves that the horizontal lines of this table furniſh the coefficients of the terms of the correſpondent powers of a binomial; and teaches how to uſe them in extracting the roots of all powers what⯑ever. And after him the ſame table was uſed for the ſame purpoſe, by Cardan, and Stevin, and the other writers on arithmetic. I ſuſpect, however, that the nature of this table was known much earlier than the time of Stifelius, at leaſt ſo far as regards the progreſſions of figurate numbers, a doctrine amply treated of by Nicomachus, who lived, according to ſome, before Euclid, but not till long after him according to others; and whoſe work on arithmetic was publiſhed at Paris in 1538; and which it is ſuppoſed was chiefly copied in the treatiſe on the ſame ſubject by Boethius: but I have never ſeen either [70]of theſe two works. Though indeed Cardan ſeems to aſcribe the in⯑vention of the table to Stifelius; but I ſuppoſe that is only to be underſtood of its application to the extraction of roots. See Cardan's Opus Novum de Proportionibus, where he quotes it, and extracts the table and its uſe from Stifelius's book. Cardan alſo, at page 185, et ſeq. of the ſame work, makes uſe of a like table to find the number of va⯑riations of things, or conjugations as he calls them.

8. The contemplation of this table has probably been attended with the invention and extenſion of ſome of our moſt curious diſcoveries in mathematics, both in regard to the powers of a binomial, with the conſequent extraction of roots, the doctrine of angular ſections by Vieta, and the differential method by Briggs and others. For, one or two of the powers or ſections being once known, the table would be of excel⯑lent uſe in diſcovering and conſtructing the reſt. And accordingly we find this table uſed on many occaſions by Stifelius, Cardan, Stevin, Vieta, Briggs, Oughtred, Mercator, Paſcal, &c. &c.

9. On this occaſion I cannot help mentioning the ample manner in which I ſee Stifelius, at fol. 35, et ſeq. of the ſame book, treats of the nature and uſe of logarithms, though not under the ſame name, but under the idea of a ſeries of arithmeticals, adapted to a ſeries of geo⯑metricals. He there explains all their uſes; ſuch as that the addition of them, anſwers to the multiplication of their geometricals; ſubtraction to diviſion; multiplication of exponents, to involution; and dividing of exponents, to evolution. And he exemplifies the uſe of them in caſes of the Rule-of-Three, and in finding mean proportionals between given terms, and ſuch like, exactly as is done in logarithms. So that he ſeems to have been in the full poſſeſſion of the idea of logarithms, and wanted only the neceſſity of troubleſome calculations to induce him to make a table of ſuch numbers.

[71]10. But although the nature and conſtruction of this table, namely of figurate numbers, was thus early known, and employed in the raiſing of powers, and extracting of roots; yet it was only by raiſing the num⯑bers one from another by continual additions, and then taking them from the table for uſe when wanted; till Briggs firſt pointed out the way of raiſing any horizontal line in the foregoing table by itſelf, with⯑out any of the preceding lines; and thus teaching to raiſe the terms of any power of a binomial, independent of any other powers; and ſo gave the ſubſtance of the binomial ſeries in words, wanting only the notation in ſymbols; as I have ſhewn at large at page 75 of the hiſtorical intro⯑duction to my Mathematical Tables.

11. Whatever was known however of this matter, related only to pure or integral powers, no one before Newton having thought of extracting roots by infinite ſeries. He happily diſcovered, that, by conſidering powers and roots in a continued ſeries, roots being as powers having fractional exponents, the ſame binomial ſeries would equally ſerve for them all, whether the index ſhould be fractional or integral, or the ſeries be finite or infinite.

12. The truth of this method however was long known only by trial in particular caſes, and by induction from analogy. Nor does it appear that even Newton himſelf ever attempted any direct proof of it. But various demonſtrations of this theorem have been ſince given by the more modern mathematicians, of which ſome are by means of the doc⯑trine of fluxions, and others, more legally, from the pure principles of algebra only. Some of which I ſhall here give a ſhort account of.

13. One of the firſt was Mr. James Bernoulli. His demonſtration is, among ſeveral other curious things, contained in his little work called Ars Conjectandi, which has been improperly omitted in the collection of his works publiſhed by his nephew Nicholas Bernoulli. This is a ſtrict [72]demonſtration of the binomial theorem in the caſe of integral and affirmative powers, and is to this effect. Suppoſing the theorem to be true in any one power, as for inſtance, in the cube, it muſt be true in the next higher power; which he demonſtrates. But it is true in the cube, in the fourth, fifth, ſixth, and ſeventh powers, as will eaſily appear by trial, that is by actually raiſing thoſe powers by continual multipli⯑cations. Therefore it is true in all higher powers. All this he ſhews in a regular and legitimate manner, from the principles of multipli⯑cation, and without the help of fluxions. But he could not extend his proof to the other caſes of the binomial theorem, in which the powers are fractional. And this demonſtration has been copied by Mr. John Stewart, in his commentary on Sir Iſaac Newton's quadrature of curves. To which he has added, from the principles of fluxions, a demonſtra⯑tion of the other caſe, for roots or fractional exponents.

14. In No. 230 of the Philoſophical Tranſactions for the year 1697, is given a theorem, by Mr. De Moivre, in imitation of the bi⯑nomial theorem, which is extended to any number of terms, and thence called the multinomial theorem; which is a general expreſſion in a ſeries, for raiſing any multinomial quantity to any power. His de⯑monſtration of the truth of this theorem, is independent of the truth of the binomial theorem, and contains in it a demonſtration of the binomial theorem as a ſubordinate propoſition, or particular caſe of the other more general theorem. And this demonſtration may be con⯑ſidered as a legitimate one, for pure powers, founded on the principles of multiplication, that is, on the doctrine of combinations and permu⯑tations. And it proves that the law of the continuation of the terms, muſt be the ſame in the terms not computed, or not ſet down, as in thoſe that are written down.

15. The ingenious Mr. Landen has given an inveſtigation of the binomial theorem, in his Diſcourſe concerning the Reſidual Analyſis, printed in 1758, and in the Reſidual Analyſis itſelf, printed in 1764. [73]The inveſtigation is deduced from this lemma, namely, if m and n be any integers, and q = v/x, then is [...] which theorem is made the principal baſis of his Reſidual Analyſis.

The inveſtigation is this: the binomial propoſed being [...], aſ⯑ſume it equal to the following ſeries 1 + ax + bx2 + cx3 &c. with indeterminate coefficients. Then for the ſame reaſon as [...] will [...] Then, by ſubtraction, [...] And, dividing both ſides by x − y, and by the lemma, we have [...] Then, as this equation muſt hold true whatever be the value of y, take y = x, and it will become [...] Conſequently, multiplying by 1 + x, we have [...], or its equal by the aſſumption, viz. [...] [...] [74]Then, by comparing the homologous terms, the value of the coefficients a, b, c, &c. are deduced for as many terms as you compare.

And a large account is given of this inveſtigation by the learned Dr. Hales, in his Analyſis Equationum, lately publiſhed at Dublin.

Mr. Landen then contraſts this inveſtigation with that by the method of fluxions, which is as follows. Aſſume as before; [...] Take the fluxion of each ſide, and we have [...] Divide by ẋ, or take it = 1, ſo ſhall [...]

Then multiply by 1 + x, and ſo on as above in the other way.

16. Beſides the above, which are the principal demonſtrations and inveſtigations that have been given of this important theorem, I have been ſhewn an ingenious attempt of Mr. Baron Maſeres, to demon⯑ſtrate this theorem in the caſe of roots or fractional exponents, by the help of De Moivre's multinomial theorem. But, not being quite ſatiſ⯑fied with his own demonſtration, as not expreſſing the law of continua⯑tion of the terms which are not actually ſet down, he was pleaſed to urge me to attempt a more complete and ſatisfactory demonſtration of the general caſe of roots, or fractional exponents. And he farther propoſed it in this form, namely, that if Q be the coefficient of one of the terms of the ſeries which is equal to [...], and P the coefficient of the next preceding term, and R the coefficient of the next follow⯑lowing term; then, if Q be [...], to prove that R will be [...]. This he obſerved would be quite perfect and ſatisfactory, [75]as it would include all the terms of the ſeries, as well thoſe that are omitted, as thoſe that are actually ſet down. And I was, in my de⯑monſtration, to ſuppoſe, if I pleaſed, the truth of the binomial and mul⯑tinomial theorems for integral powers, as truths that had been previ⯑ouſly and perfectly proved.

In conſequence I ſent him ſoon after the ſubſtance of the following demonſtration; with which he was quite ſatisfied, and which I now proceed to explain at large.

17. Now the binomial integral is [...]. where a, b, c, &c. denote the whole coefficients of the 2d, 3d, 4th, &c. terms, over which they are placed; and in which the law is this, namely, if P, Q, R, be the coefficients of any three terms in ſucceſſion, and if g/b P = Q, then is [...]; as is evident; and which, it is granted, has been proved.

18. And the binomial fractional is [...]. in which the law is this, namely, if P, Q, R be the coefficients of three terms in ſucceſſion; and if g/b P = Q, then is [...]. Which is the property to be proved.

[76]19. Again, the multinomial integral is [...] [77] [...] &c. Or, if we put a, b, c, d, &c. for the coefficients of the 2d, 3d, 4th, 5th, &c. terms, the laſt ſeries, by ſubſtitution, will be transformed into this form, [...]

[78]20. Now, to find the ſeries in Art. 18, aſſume the propoſed binomial equal to a ſeries with indeterminate coefficients, as [...] Then raiſe each ſide to the n power, ſo ſhall [...]. But it is granted that the multinomial raiſed to any integral power is proved, and known to be, as in the laſt Art. [...] It follows then, that if this laſt ſeries be equal to 1 + x, by equating the homologous coefficients, all the terms after the ſecond muſt vaniſh, or all the coefficients b, c, d, &c. after the ſecond term, muſt be each = 0. Writing therefore, in this ſeries, 0 for each of the letters b, c, d, &c. it will become of this more ſimple form, [...]. Put now each of the coefficients, after the ſecond term, = 0, and we ſhall have theſe equations [...] [...] [...] [...] &c. [79]The reſolution of which equations gives the following values of the aſſumed indeterminate coefficients, namely, [...], &c. which coefficients are according to the law propoſed, namely, when g/h P = Q, then g−n/h+n Q = R. Q. E. D.

21. Alſo, by equating the ſecond coefficients, namely, 1 = a = nA, we find A = 1/n. This being written for A in the above values of B, C, D, &c. will give the proper ſeries for the binomial in queſtion, namely [...].

TRACT VII. Of the Common Sections of the Sphere and Cone. Together with the Demonſtration of ſome other New Properties of the Sphere, which are ſimilar to certain Known Properties of the Circle.

THE ſtudy of the mathematical ſciences is uſeful and profitable, not only on account of the benefit derivable from them to the affairs of mankind in general; but are moſt eminently ſo, for the plea⯑ſure and delight the human mind feels in the diſcovery and contempla⯑tion of the endleſs number of truths that are continually preſenting themſelves to our view. Theſe meditations are of a ſublimity far above all others, whether they be purely intellectual, or whether they reſpect the nature and properties of material objects: they methodiſe, ſtrengthen, and extend the reaſoning faculties in the moſt eminent de⯑gree, and ſo fit the mind the better for underſtanding and improving every other ſcience; but, above all, they furniſh us with the pureſt and moſt permanent delight, from the contemplation of truths peculiarly certain and immutable, and from the beautiful analogy which reigns through all the objects of ſimilar inquiry. In the mathematical ſciences, the diſcovery, often accidental, of a plain and ſimple property, is but the harbinger of a thouſand others of the moſt ſublime and beautiful nature, to which we are gradually led, delighted, from the more ſimple to the more compound and general, till the mind becomes quite en⯑raptured at the full blaze of light burſting upon it from all directions.

[84]Of theſe very pleaſing ſubjects, the ſtriking analogy that prevails among the properties of geometrical figures, or figured extenſion, is not one of the leaſt. Here we often find that a plain and obvious property of one of the ſimpleſt figures, leads us to, and forms only a particular caſe of, a property in ſome other figure, leſs ſimple; afterwards this again turns out to be no more than a particular caſe of another ſtill more general; and ſo on, till at laſt we often trace the tendency to end in a general property of all figures whatever.

The few properties which make a part of this paper, conſtitute a ſmall ſpecimen of the analogy, and even identity, of ſome of the more remarkable properties of the circle, with thoſe of the ſphere. To which are added ſome properties of the lines of ſection, and of contact, be⯑tween the ſphere and cone. Both which may be farther extended as occaſions may offer: like as all of theſe properties have occurred from the circumſtance, mentioned near the end of the paper, of conſidering the inner ſurface of a hollow ſpherical veſſel, as viewed by an eye, or as illuminated by rays, from a given point.

TRACT VIII. Of the Geometrical Diviſion of Circles and Ellipſes into any Number of Parts, and in any propoſed Ratios.

ART. 1. IN the year 1774 was publiſhed a pamphlet in octavo, with this title, A Diſſertation on the Geometrical Analyſis of the Antients. With a Collection of Theorems and Problems, without Solu⯑tions, for the Exerciſe of Young Students. This pamphlet was anony⯑mous; it was however well known to myſelf and ſeveral other perſons, that the author of it was the late Mr. John Lawſon, B. D. rector of Swanſcombe in Kent, an ingenious and learned geometrician, and, what is ſtill more eſtimable, a moſt worthy and good man; one in whoſe heart was found no guile, and whoſe pure integrity, joined to the moſt amiable ſimplicity of manners, and ſweetneſs of temper, gained him the affection and reſpect of all who had the happineſs to be acquainted with him. His collection of problems in that pamphlet concluded with this ſingular one, "To divide a circle into any number of parts, which ſhall be as well equal in area as in circumference.—N. B. This may ſeem a paradox, however it may be effected in a manner ſtrictly geo⯑metrical." The ſolution of this ſeeming paradox he reſerved to him⯑ſelf, as far as I know. I fell upon the diſcovery however ſoon after; and other perſons might do the ſame. My reſolution of it was pub⯑liſhed in an account which I gave of the pamphlet in the Critical Review for 1775, vol. xl. and which the author informed me was on [94]the ſame principle as his own. This account is in page 21 of that volume, and in the following words:

2. "We have no doubt but that our mathematical readers will agree with us in allowing the truth of the author's remark concerning the ſeeming paradox of this problem; becauſe there is no geometrical method of dividing the circumference of a circle into any propoſed number of parts taken at pleaſure, and it does not readily appear that there can be any othermethod of reſolving the problem, than by draw⯑ing radii to the points of equal diviſion in the circumference. However another method there is, and that ſtrictly geometrical, which is as follows.

"Divide the diameter AB of the given circle into as many equal parts as the circle itſelf is to be divided into, at the points C, D, E, &c. Then on the diameters AC, AD, AE, &c. as alſo on BE, BD, BC, &c. deſcribe ſemicircles, as in the annexed figure: and they will divide the whole circle as required.

"For, the ſeveral diameters being in arithmetical progreſſion, of which the common difference is equal to the leaſt of them, and the dia⯑meters of circles being as their circumferences, theſe will alſo be in arithmetical progreſſion. But, in ſuch a progreſſion, the ſum of the ex⯑tremes is equal to the ſum of each two terms equally diſtant from them; therefore the ſum of the circumferences on AC and CB, is equal to the ſum of thoſe on AD and DB, and of thoſe on AE and EB, &c. and each ſum equal to the ſemi-circumference of the given circle on the diameter AB. Therefore all the parts have equal perimeters, and each is equal to the circumference of the propoſed circle. Which ſatisfies one of the conditions in the problem.

[95]"Again, the ſame diameters being as the numbers 1, 2, 3, 4, &c. and the areas of circles being as the ſquares of their diameters, the ſemicircles will be as the numbers 1, 4, 9, 16, &c. and conſequently the differences between all the adjacent ſemicircles are as the terms of the arithmetical progreſſion 1, 3, 5, 7, &c. and here again the ſums of the extremes, and of every two equidiſtant means, make up the ſeve⯑ral equal parts of the circle. Which is the other condition."

3. But this ſubject admits of a more geometrical form, and is capa⯑ble of being rendered very general and extenſive, and is moreover very fruitful in curious conſequences. For firſt, in whatever ratio the whole diameter is divided, whether into equal or unequal parts, and whatever be the number of the parts, the peri⯑meters of the ſpaces will ſtill be equal. For ſince circumferences of circles are always as their diameters, and becauſe AB and AD + DB and AC + CB are all equal, therefore the ſemi-circum⯑ferences c and b + d and a + e are all equal, and conſtant, whatever be the ratio of the parts AD, DC, CB, of the diame⯑ter. We ſhall preſently find too that the ſpaces TV, RS, and PQ, will be univerſally as the ſame parts AD, DC, CB, of the diameter.

4. The ſemicircles having been deſcribed as before mentioned, erect CE perpendicular to AB, and join BE. Then I ſay, the circle on the diameter BE, will be equal to the ſpace PQ. For, join AE. [96]Now the ſpace P = ſemicircle on AB − ſemicircle on AC: but the ſemicir. on AB = ſemicir. on AE + ſemicir. on BE, and the ſemicir. on AC = ſemicir. on AE − ſemicir. on CE, theref. ſemic. AB − ſemic. AC = ſemic. BE + ſemicir. CE, that is the ſpace P is = ſemic. BE + ſemicir. CE; to each of theſe add the ſpace Q, or the ſemicircle on BC, then P + Q = ſemic. BE + ſemic. CE + ſemic. BC, that is P + Q = double the ſemic. BE, or = the whole circle on BE.

5. In like manner, the two ſpaces PQ and RS together, or the whole ſpace PQRS, is equal to the circle on the diameter BF. And therefore the ſpace RS alone, is equal to the difference, or the circle on BF minus the circle on BE.

6. But, circles being as the ſquares of their diameters, BE², BF², and theſe again being as the parts or lines BC, BD, therefore the ſpaces PQ, PQRS, RS, TV, are reſpectively as the lines BC, BD, CD, AD, And if BC be equal to CD, then will PQ be equal to RS, as in the firſt or ſimpleſt caſe.

7. Hence, to find a circle equal to the ſpace RS, where the points D and C are taken at random: From either end of the diameter, as A, take AG equal to DC, erect GH perpendicular to AB, and join AH; then the circle on AH will be equal to the ſpace RS. For, the ſpace PQ: the ſpace RS ∷ BC ∶ CD or AG, that is as BE²: AH² the ſquares of the diameters, or as the circle on BE to the cir⯑cle on AH; but the circle on BE is equal to the ſpace PQ, and therefore the circle on AH is equal to the ſpace RS.

8. Hence, to divide a circle in this manner, into any number of parts, that ſhall be in any ratios to one another: Divide the diameter [97]into as many parts, at the points D, C, &c. and in the ſame ratios as thoſe propoſed; then on the ſeveral diſtances of theſe points from the two ends A and B, as diameters, deſcribe the alternate ſemicircles on the different ſides of the whole diameter AB: and they will divide the whole circle in the manner propoſed. That is, the ſpaces TV, RS, PQ, will be as the lines AD, DC, CB.

9. But theſe properties are not confined to the circle alone, but are to be found alſo in the ellipſe, as the genus of which the circle is only a ſpecies. For if the annexed figure be an ellipſe deſcribed on the axis AB, the area of which is, in like manner, divided by ſimilar ſemiellipſes, deſcribed on AD, AC, BC, BD, as axes, all the ſemiperimeters f, ae, bd, c, will be equal to one another, for the ſame reaſon as before in Art. 3, namely, becauſe the peripheries of ellipſes are as their diameters. And the ſame property would ſtill hold good, if AB were any other diameter of the ellipſe, inſtead of the axis; deſcribing upon the parts of it ſemiellipſes which ſhall be ſimilar to thoſe into which the diameter AB divides the given ellipſe.

10. And, if a circle be deſcribed about the ellipſe, on the diame⯑ter AB, and lines be drawn ſimilar to thoſe in the ſecond figure; then, by a proceſs the very ſame as in Art. 4, et ſeq. ſubſtituting only ſemiellipſe for ſemicircle, it is found that the ſpace

• PQ is equal to the ſimilar ellipſe on the diameter BE,
• PQRS is equal to the ſimilar ellipſe on the diameter BF,
• RS is equal to the ſimilar ellipſe on the diameter AH,

or to the difference of the ellipſes on BF and BE; alſo the elliptic ſpaces PQ, PQRS, RS, TV, are reſpectively as the lines BC, BD, DC, AD, [98]the ſame ratio as the circular ſpaces. And hence an ellipſe is divided into any number of parts, in any aſſigned ratios, in the ſame manner as the circle is divided, namely, dividing the axis, or any diameter in the ſame manner, and on the parts deſcribing ſimilar ſemi⯑ellipſes.

TRACT IX. New Experiments in Artillery; for determining the Force of fired Gun⯑powder, the Initial Velocity of Cannon Balls, the Ranges of Pieces of Cannon at different Elevations, the Reſiſtance of the Air to Projectiles, the Effect of different Lengths of Cannon, and of different Quantities of Powder, &c. &c.

Sect. 1. AT Woolwich in the year 1775, in conjunction with ſome able officers of the Royal Regiment of Artillery, and other ingenious gentlemen, I firſt inſtituted a courſe of experiments on fired gunpowder and cannon balls. My account of them was preſented to the Royal Society, who honoured it with the gift of the annual gold medal, and printed it in the Philoſophical Tranſactions for the year 1778. The object of thoſe experiments, was the determination of the actual velocities with which balls are impelled from given pieces of cannon, when fired with given charges of powder. They were made according to the method invented by the very ingenious Mr. Robins, and de⯑ſcribed in his treatiſe on the new principles of gunnery, of which an account was printed in the Philoſophical Tranſactions for the year 1743. Before the diſcoveries and inventions of that gentleman, very little progreſs had been made in the true theory of military projectiles. His book however contained ſuch important diſcoveries, that it was ſoon tranſlated into ſeveral of the languages on the continent, and the late ſamous Mr. L. Euler honoured it with a very learned and extenſive commentary, in his tranſlation of it into the German language. That [100]part of Mr. Robins's book has always been much admired, which re⯑lates to the experimental method of aſcertaining the actual velocities of ſhot, and in imitation of which, but on a large ſcale, thoſe experiments were made which were deſcribed in my paper. Experiments in the manner of Mr. Robins were generally repeated by his commentators, and others, with univerſal ſatisfaction; the method being ſo juſt in theory, ſo ſimple in practice, and altogether ſo ingenious, that it imme⯑diately gave the fulleſt conviction of its excellence, and the eminent abilities of the inventor. The uſe which our author made of his inven⯑tion, was to obtain the real velocities of bullets experimentally, that he might compare them with thoſe which he had computed a priori from a new theory of gunnery which he had invented, in order to verify the principles on which it was founded. The ſucceſs was fully anſwerable to his expectations, and left no doubt of the truth of his theory, at leaſt when applied to ſuch pieces and bullets as he had uſed. Theſe however were but ſmall, being only muſket balls of about an ounce weight: for, on account of the great ſize of the machinery neceſſary for ſuch experiments, Mr. Robins, and other ingenious gentlemen, have not ventured to ex⯑tend their practice beyond bullets of that kind, but contented them⯑ſelves with ardently wiſhing for experiments to be made in a ſimilar manner with balls of a larger ſort. By the experiments deſcribed in my paper therefore I endeavoured, in ſome degree, to ſupply that de⯑fect, having uſed cannon balls of above twenty times the ſize, or from one pound to near three pounds weight. Thoſe are the only experi⯑ments, that I know of, which have been made in that way with can⯑non balls, although the concluſions to be deduced from ſuch a courſe, are of the greateſt importance in thoſe parts of natural philoſophy which are connected with the effects of fired gunpowder: nor do I know of any other practical method beſides that above, of aſcertaining the initial velocities of military projectiles within any tolerable degree of the truth; except that of the recoil of the gun, hung on an axis in the ſame manner as the pendulum; which was alſo firſt pointed out and uſed by Mr. Robins, and which has lately been practiſed alſo by Benjamin [101]Thompſon, Eſq. in his very ingenious and accurate ſet of experiments with muſket balls, deſcribed in his paper in the Philoſophical Tranſ⯑actions for the year 1781. The knowledge of this velocity is of the greateſt conſequence in gunnery: by means of it, together with the law of the reſiſtance of the medium, every thing is determinable which re⯑lates to that buſineſs; for, as I remarked in the paper above-mentioned on my firſt experiments, it gives us the law relative to the different quantities of powder, to the different weights of balls, and to the dif⯑ferent lengths and ſizes of guns, and it is alſo an excellent method of trying the ſtrength of different ſorts of powder. Beſide theſe, there does not ſeem to be any thing wanting to anſwer every inquiry that can be made concerning the flight and ranges of ſhot, except the effects ariſing from the reſiſtance of the medium.

2. In that courſe of experiments were compared the effects of differ⯑ent quantities of powder, from two to eight ounces; the effects of different weights of ſhot; and the effects of different ſizes of ſhot, or different degrees of windage, which is the difference between the dia⯑meter of the ſhot and the diameter of the bore; all of which were found to obſerve certain regular and conſtant laws, as far as the experi⯑ments were carried. And at the end of each day's experiments, the deductions and concluſions were made, and the reaſons clearly pointed out why ſome caſes of velocity differ from others, as they properly and regularly ought to do. So that I am ſurprized how they could be miſ⯑underſtood by Mr. Templehof, captain in the Pruſſian artillery, when ſpeaking of the irregularities in ſuch experiments, he ſays, (page 126 of Le Bombardier Pruſſien, printed at Berlin, 1781) "La meme choſe arriva a Mr. Hutton, il la trouva de 626 pieds, & le jour ſuivant de 973 pieds, tout les circonſtances étant d'ailleurs égales:" which laſt words ſhew that Mr. T. had either miſunderſtood, or had not read the reaſon, which is a very ſufficient one, for this remarkable difference: it is expreſsly remarked in page 71 of my paper in the Philoſophical [102]Tranſactions, that all the circumſtances were not the ſame, but that the one ball was much ſmaller than the other, and that it had the leſs de⯑gree of velocity, 626 feet, becauſe of the greater loſs of the elaſtic fluid by the windage in the caſe of the ſmaller ball. On the contrary, the velocities in thoſe experiments were even more uniform and ſimilar thancould be expected in ſuch large machinery, and in a firſt attempt of the kind too. And from the whole, the following important con⯑cluſions were fairly drawn and ſtated, viz.

"(1.) And firſt, it is made evident by theſe experiments, that pow⯑der fires almoſt inſtantaneouſly, ſeeing that almoſt the whole of the charge fires, though the time be much diminiſhed.

"(2.) The velocities communicated to ſhot of the ſame weight, with different quantities of powder, are nearly in the ſubduplicate ratio of thoſe quantities. A very ſmall variation, in defect, taking place when the quantities of powder become great.

"(3.) And when ſhot of different weights are fired with the ſame quantity of powder, the velocities communicated to them, are nearly in the reciprocal ſub-duplicate ratio of their weights.

"(4.) So that, univerſally, ſhot which are of different weights, and impelled by the firing of different quantities of powder, acquire velo⯑cities which are directly as the ſquare roots of the quantities of powder, and inverſely as the ſquare roots of the weights of the ſhot, nearly.

"(5.) It would therefore be a great improvement in artillery, to make uſe of ſhot of a long form, or of heavier matter; for thus the mo⯑mentum of a ſhot, when fired with the ſame weight of powder, would be increaſed in the ratio of the ſquare root of the weight of the ſhot.

[103]"(6.) It would alſo be an improvement, to diminiſh the windage: for, by ſo doing, one third or more of the quantity of powder might be ſaved.

"(7.) When the improvements mentioned in the laſt two articles are conſidered as both taking place, it is evident that about half the quantity of powder might be ſaved; which is a very conſiderable ob⯑ject. But important as this ſaving may be, it ſeems to be ſtill ex⯑ceeded by that of the guns: for thus a ſmall gun may be made to have the effect and execution of one of two or three times its ſize in the preſent way, by diſcharging a long ſhot of two or three times the weight of its natural ball, or round ſhot: and thus a ſmall ſhip might diſcharge ſhot as heavy as thoſe of the greateſt now made uſe of.

"Finally, as the above experiments exhibit the regulations with re⯑gard to the weight of powder and balls, when fired from the ſame piece of ordnance; ſo by making ſimilar experiments with a gun, varied in its length, by cutting off from it a certain part before each courſe of experiments, the effects and general rules for the different lengths of guns, may be certainly determined by them. In ſhort, the principles on which theſe experiments were made, are ſo fruitful in conſequences, that, in conjunction with the effects of the reſiſtance of the medium, they ſeem to be ſufficient for anſwering all the inquiries of the ſpeculative philoſopher, as well as thoſe of the practical artilleriſt."

3. Such then was the ſtate of the firſt ſet of experiments with cannon balls in the year 1775, and ſuch were the probable advantages to be derived from them. I do not however know that any uſe has hitherto been made of them by authority for the public ſervice; unleſs perhaps we are to except the inſtance of Carronades, a ſpecies of ordnance which hath ſince been invented, and in ſome degree adopted in the public [104]ſervice; for in this inſtance the proprietors of thoſe pieces, by availing themſelves of the circumſtances of large balls, and very ſmall windage, with ſmall charges of powder, have been able to produce very conſider⯑able and uſeful effects with thoſe light pieces, at a very ſmall expence. Or perhaps thoſe experiments were too much limited, and of too pri⯑vate a nature, to merit a more general notice. Be that however as it may, the preſent additional courſe, which is to make the ſubject of this tract, will have very great advantages over the former, both in point of extent, variety, improvements in machinery, and in authority. His Grace the Duke of Richmond, the preſent maſter-general of the ordnance, in his indefatigable endeavours for the good of the public ſervice, was pleaſed to order this extenſive courſe of experiments, and to give directions for providing guns, and machinery, and every thing compleat and fitting for the proper execution of them.

4. This courſe of experiments has been carried on under the direction of Major Blomefield, inſpector of artillery, an officer of great profeſ⯑ſional merit, and whoſe ingenious contrivances in the machinery do him great credit. It has been our employment for three ſucceſſive ſummers, namely, thoſe of the years 1783, 1784, and 1785; and indeed it might be continued ſtill much longer, either by extending it to more objects, or to more repetitions of experiments for the ſame object.

5. The objects of this courſe have been various. But the principal articles of it as follows:

• (1.) The velocities with which balls are projected by equal charges of powder, from pieces of the ſame weight and calibre, but of different lengths.
• (2.) The velocities with different charges of powder, the weight and length of the gun being the ſame.
• [105](3.) The greateſt velocity due to the different lengths of guns, to be obtained by increaſing the charge as far as the reſiſtance of the piece is capable of ſuſtaining.
• (4.) The effect of varying the weight of the piece; every thing elſe being the ſame.
• (5.) The penetration of balls into blocks of wood.
• (6.) The ranges and times of flight of balls; to compare them with their initial velocities, for determining the reſiſtance of the medium.
• (7.) The effect of wads; of different degrees of ramming, or compreſſing the charge; of different degrees of windage; of different poſitions of the vent; of chambers, and trunnions, and every other circumſtance neceſſary to be known for the improvement of artillery.