CHAPTER IV. ARITHMETIC IN ARCHIMEDES. Two of the treatises, the Measurement of a circle and the Sand-reckoner, are mostly arithmetical in content. Of the Sand-reckoner nothing need be said here, because the system for expressing numbers of any magnitude which it unfolds and applies cannot be better described than in the book itself; in the Measurement of a circle, however, which involves a great deal of manipulation of numbers of considerable size though expressible by means of the ordinary Greek notation for numerals, Archimedes merely gives the results of the various arithmetical operations, multiplication, extraction of the square root, etc., without setting out any of the operations themselves. Various interesting questions are accordingly involved, and, for the convenience of the reader, I shall first give a short account of the Greek system of numerals and of the methods by which other Greek mathematicians usually performed the various operations included under the general term XoyurTuaj (the art of calculating), in order to lead up to an explanation (1) of the way in which Archimedes worked out approximations to the square roots of large numbers, (2) of his method of arriving at the two approximate values of which he simply sets down without any hint as to how they were obtained*. * In writing this chapter I have been under particular obligations to Hultsch's articles Arithmetiea and Archimedes in Pauly-Wissowa's Beal-Encyclopädie, n. 1, as well as to the same scholar's articles (1) Die Näherung swerthe irrationaler Quadratwurzeln bei Archimedes in the Nachrichten von der kgl. Gesellschaft der Wissenschaften zu Güttingen (1893), pp. 367 sqq., and (2) Zur Kreismessung des Archimedes in the Zeitschrift für Math. u. Physik (Hist. litt. Abtheilung) xxxix. (1894), pp. 121 sqq. and 161 sqq. I have also made use, in the earlier part of the chapter, of Nesselmann's work Die Algebra der Griechen and the histories of Cantor and Gow. ARITHMETIC IN ARCHIMEDES. lxix § 1. Greek numeral system. It is well known that the Greeks expressed all numbers from 1 to 999 by means of the letters of the alphabet reinforced by the addition of three other signs, according to the following scheme, in which however the accent on each letter might be replaced by a short horizontal stroke above it, as a. a, p, y, 8', t, r', C, y, 0' are 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. i't k, X', M'f v\ o', q' „ 10, 20, 30, ......... 90 p', 8«rj(tXioi and so on). Various abbreviations were used for the word pvpias, the most common being M or Mi; and, where this was used, the number of myriads, or the multiple of 10,000, was generally written over the abbreviation, though some- times before it and even after it. Thus 349,450 was M8vv*. Fractions (kerra) were written in a variety of ways. The most usual was to express the denominator by the ordinary numeral with two accents affixed. When the numerator was unity, and it was therefore simply a question of a symbol for a single word such as * Diophantus denoted myriads followed by thousands by the ordinary signs for numbers of units, only separating them by a dot from the thousands. Thus for 3,069,000 he writes Tr.fi, and Ay . ra\j/m for 331,776. Sometimes myriads were represented by the ordinary letters with two dots above, as j> = 100 myriads (1,000,000), and myriads of myriads with two pairs of dots, as 1' for 10 myriad-myriads (1,000,000,000). lxx INTRODUCTION. Tpirov, ^, there was no need to express the numerator, and the symbol was y"; similarly r" = J, if." = ^r, and so on. "When the numerator was not unity and a certain number of fourths, fifths, etc., had to be expressed, the ordinary numeral was used for the numerator; thus & ta" = T9T, i oa." = iJ. In Heron's Geometry the denominator was written twice in the latter class of fractions; thus \ (hio Tri/jLiTTa) was ySY'e", ^| (Aeirra TpiaKOOToYpiTa Ky' or UKOtJiTpia TpiaKotJTOTpiTa) was Ky Xy" Ay". The sign for ^, tffLum, is in Archimedes, Diophantus and Eutocius L", in Heron C or a sign similar to a capital S*. A favourite way of expressing fractions with numerators greater than unity was to separate them into component fractions with numerator unity, when juxtaposition as usual meant addition. Thus | was written L'T = $ + \; if- was CS'VV = \ + \ + I + TV ; Eutocius writes U"|8" or | + ^ for ff, and so on. Sometimes the same fraction was separated into several different sums; thus in Heron (p. 119, ed. Hultsch) -||~| is variously expressed as (a) J + \ + + T^2- + zhr, and (o) ^+1+^- + ^ + -^. Sexagesimal fractions. This system has to be mentioned because the only instances of the working out of some arithmetical operations which have been handed down to us are calculations expressed in terms of such fractions; and moreover they are of special interest as having much in common with the modern system of decimal fractions, with the difference of course that the submultiple is 60 instead of 10. The scheme of sexagesimal fractions was used by the Greeks in astronomical calculations and appears fully developed in the awTa^is of Ptolemy. The circumference of a circle, and along with it the four right angles subtended by it at the centre, are divided into 360 parts (iy«f/u,aTa or fioipai) or as we should say degrees, each poipa into 60 parts called (first) sixtieths, (irpona) IfijKoo-ra, or minutes (Xeirrd), each of these again into Stvrepa If^Koerra (seconds), and so on. A similar division of the radius of the circle into 60 * Diophantus has a general method of expressing fractions which is the exact reverse of modern practice; the denominator is written above the 7 Ke a. oils- numerator, thus 7=5/3, /co = 21/25, and picf. 0£j =1,270,568/10,816. Sometimes he writes down the numerator and then introduces the denominator with in fwply or fioplov, e.g. rr. fiop. \y.a\j/x = 3,069,000/331,776. ARITHMETIC IN ARCHIMEDES. lxxi parts (T/x-ijiiaTa) was also made, and these were each subdivided into sixtieths, and so on. Thus a convenient fractional system was available for general arithmetical calculations, expressed in units of any magnitude or character, so many of the fractions which we should represent by so many of those which we should write (fV)2' (bV)3> an<^ so on ^° anv extent. It is therefore not surprising that Ptolemy should say in one place " In general we shall use the method of numbers according to the sexagesimal manner because of the inconvenience of the [ordinary] fractions." For it is clear that the successive submultiples by 60 formed a sort of frame with fixed compartments into which any fractions whatever could be located, and it is easy to see that e.g. in additions and subtractions the sexagesimal fractions were almost as easy to work with as decimals are now, 60 units of one denomination being equal to one unit of the next higher denomination, and "carrying" and "borrowing" being no less simple than it is when the number of units of one denomination necessary to make one of the next higher is 10 instead of 60. In expressing the units of the circumference, degrees, fioipat or the symbol /J was generally used along with the ordinary numeral which had a stroke above it; minutes, seconds, etc. were expressed by one, two, etc. accents affixed to the numerals. Thus /tj5 = 2°, ixoipwv pi ft." = 47° 42' 40". Also where there was no unit in any particular denomination O was used, signifying oiBe/xia jxolpa, ovSev eirjKoo-Tov and the like ; thus O a /3" 0"' = 0° 1' 2" 0"'. Similarly, for the units representing the divisions of the radius the word t/xruxara or some equivalent was used, and the fractions were represented as before; thus rfxr^xAriav ££ 8've" = 67 (units) 4' 55". § 2. Addition and Subtraction. There is no doubt that, in writing down numbers for these purposes, the several powers of 10 were kept separate in a manner corresponding practically to our system of numerals, and the hundreds, thousands, etc., were written in separate vertical rows. The following would therefore be a typical form of a sum in addition; av k 8' = 1424 p y 103 12281 V M A. 30030 43838 lxxii INTRODUCTION. and the mental part of the work would be the same for the Greek as for us. Similarly a subtraction would be represented as follows: MyxXr' = 93636 M/7v 6' 23409 1 M <™£' 70227 § 3. Multiplication. A number of instances are given in Eutocius' commentary on the Measurement of a circle, and the similarity to our procedure is just as marked as in the above cases of addition and subtraction. The multiplicand is written first, and below it the multiplier preceded by im (= "into"). Then the highest power of 10 in the multiplier is taken and multiplied into the terms containing the separate multiples of the successive powers of 10, beginning with the highest and descending to the lowest; after which the next highest power of 10 in the multiplier is multiplied into the various denominations in the multiplicand in the same order. The same procedure is followed where either or both of the numbers to be multiplied contain fractions. Two instances from Eutocius are appended from which the whole procedure will be understood. (1) i/rir' 780 £7Tl \f/ir' X 780 MM;' 490000 56000 M,rW 56000 6400 t ojuoB M17«' sum 608400 (2) yiy' L"8" 3013Ji [=3013f] liri tyiy L"8" x 3013J^ 9,000,000 30,000 9,000 MpXe'P' L" 30,000 100 30 JSXffa' L" L"8" 9,000 30 9 j f r f \ rr^ff rr a

an(* tnis is equal to 11.602-2.67.4.60-16 _ 7424 602 " 608 ' / 4 \ y . 7424 Thus we must suppose that 2 I 67 + ^1 approximates to -qqt > or that 80 482/ is approximately equal to 7424. 60. ARITHMETIC IN ARCHIMEDES. lxxvii Therefore y is approximately equal to 55. We have then to subtract 4\55 /55V 442640 3025 2 l67 + 60j W + W) ' °r -W~ + W from the remainder -|^r above found. ™ , . . . 442640 . 7424 2800 46 40 The subtraction of from gives ""go5"' or 60* + 60s ' 3025 but Theon does not go further and subtract the remaining , 55 instead of which he merely remarks that the square of ^ 46 40 approximates to — + FKS. As a matter of fact, if we deduct the bO oO 3025 . 2800 , . , . , . . ~qqT irom -g-Qj-, so as to obtain the correct remainder, it is found to be ^f^J^. 604 To show the power of this method of extracting square roots by means of sexagesimal fractions, it is only necessary to mention that mi .103 55 23 . ±*tolemy gives -^j- +-^7r„ + ^5 as an approximation to V3, which oO oO bU approximation is equivalent to 1'7320509 in the ordinary decimal notation and is therefore correct to 6 places. But it is now time to pass to the question how Archimedes obtained the two approximations to the value of »/3 which he assumes in the Measurement of a circle. In dealing with this subject I shall follow the historical method of explanation adopted by Hultsch, in preference to any of the mostly a priori theories which the ingenuity of a multitude of writers has devised at different times. § 6. Early investigations of surds or incommensurables. From a passage in Proclus' commentary on Eucl. i.* we learn that it was Pythagoras who discovered the theory of irrationals {rj tZv aXoyav 7rpa-y/u.aTcia). Further Plato says (Theaetetus 147 d), "On square roots this Theodorus [of Cyrene] wrote a work in * p. 65 (ed. Friedlein). lxxviii INTRODUCTION. which he proved to us, with reference to those of 3 or 5 [square] feet that they are incommensurable in length with the side of one square foot, and proceeded similarly to select, one by one, each [of the other incommensurable roots] as far as the root of 17 square feet, beyond which for some reason he did not go." The reason why -J 2 is not mentioned as an incommensurable square root must be, as Cantor says, that it was before known to be such. We may therefore conclude that it was the square root of 2 which was geometrically constructed by Pythagoras and proved to be incommensurable with the side of a square in which it represented the diagonal. A clue to the method by which Pythagoras investigated the value of *J2 is found by Cantor and Hultsch in the famous passage of Plato (Rep. viii. 546 b, o) about the ' geometrical' or ' nuptial' number. Thus, when Plato contrasts the prtj-rq and apprfros Sia/uurpos t^s 7re/u.7raSos, he is referring to the diagonal of a square whose side contains five units of length ; the apprjros Siajuerpos, or the irrational diagonal, is then J50 itself, and the nearest rational number is \/50 — 1, which is the prynj Sidperpos. We have herein the explanation of the way in which Pythagoras must have made the first and most readily comprehensible approximation to si 2; he must have taken, instead of 2, an improper fraction equal to it but such that the denominator was a square in any case, while the numerator was as near as possible to a complete square. Thus Pythagoras chose , and the first approximation to J2 was 7 — 7 accordingly g, it being moreover obvious that sl2> — . Again, Pythagoras cannot have been unaware of the truth of the proposition, proved in Eucl. ii. 4, that (a + 6)2 = a2 + 2ab + b", where a, b are any two straight lines, for this proposition depends solely upon propositions in Book i. which precede the Pythagorean proposition i. 47 and which, as the basis of i. 47, must necessarily have been in substance known to its author. A slightly different geometrical proof would give the formula (a - Vf = a2 — 2ab + b°, which must have been equally well known to Pythagoras. It could not therefore have escaped the discoverer of the first approximation ij5Q — 1 for V50 that the use of the formula with the positive sign would give a much nearer approximation, viz. 7 + ^, which is only ARITHMETIC IN ARCHIMEDES. lxxix greater than J50 to the extent of (j^J • Thus we may properly assign to Pythagoras the discovery of the fact represented by 7tt>n/50>7. 14: The consequential result that J2> ~ J50 — 1 is used by Aristarchus of Samos in the 7th proposition of his work On the size and distances of the sun and moon*. With reference to the investigations of the values of J3, *J5, nj6,......«/l7 by Theodorus, it is pretty certain that «/3 was geometrically represented by him, in the same way as it appears * Part of the proof of this proposition was a sort of foretaste of the first part of Prop. 3 of Archimedes' Measurement of a circle, and the substance of it is accordingly appended as reproduced by Hultsch. ABEK is a square, KB a diagonal, L HBE = \L KBE, L FBE = 3°, and AC is perpendicular to BF so that the triangles ACB, BEF are similar. Aristarchus seeks to prove that AB : BC > 18 : 1. If R denote a right angle, the angles KBE, HBE, FBE are respectively %%R, V%B. Then HE : FE =» L HBE : L FBE. [This is assumed as a known lemma by Aristarchus as well as Archimedes.] Therefore HE : FE > 15 : 2.................................(a). Now, by construction, BIC2 = 2BE'i. Also [Eucl. vi. 3] whence And, since BK : BE = KH : HE ; KH=>J2HE. v/2 ■J- 50-1 25 ' 5, KH :HE>'! so that KE : EH > 12 : 5 ......... Prom (a) and (|3), ex aequali, KE : FE > 18 : 1. Therefore, since BF > BE (or KE), BF : FE > 18 : 1, so that, by similar triangles, AB : BC > 18 : 1. lxxx INTRODUCTION. afterwards in. Archimedes, as the perpendicular from an angular point of an equilateral triangle on the opposite side. It would thus be readily comparable with the side of the "1 square foot" mentioned by Plato. The fact also that it is the side of three square feet (tpiVovs SvVa/us) which was proved to be incommensurable suggests that there was some special reason in Theodorus' proof for specifying feet, instead of units of length simply; and the explanation is probably that Theodorus subdivided the sides of his triangles in the same way as the Greek foot was divided into halves, fourths, eighths and sixteenths. Presumably therefore, exactly as Pythagoras had approximated to J2 by putting ^ 48 for 2, Theodorus started from the identity 3 = . It would then be clear that To investigate V48 further, Theodorus would put it in the form \/49 — 1, as Pythagoras put V'50 into the form \/49 + l, and the result would be VF8(= v/493i)<7-i. We know of no further investigations into incommensurable square roots until we come to Archimedes. § 7. Archimedes' approximations to *J3. Seeing that Aristarchus of Santos was still content to use the first and very rough approximation to \/2 discovered by Pythagoras, it is all the more astounding that Aristarchus' younger contemporary Archimedes should all at once, without a word of explanation, give out that 1351 l7i 265 W>V3>153' as he does in the Measurement of a circle. In order to lead up to the explanation of the probable steps by which Archimedes obtained these approximations, Hultsch adopts the same method of analysis as was used by the Greek geometers in solving problems, the method, that is, of supposing the problem solved and following out the necessary consequences. To compare ARITHMETIC IN ARCHIMEDES. lxxxi 265 1351 the two fractions and -jgQ > we nrst divide both denominators into their smallest factors, and we obtain 780= 2.2.3.5.13, 153 = 3.3.17. We observe also that 2.2.13 = 52, while 3.17 = 51, and we may therefore show the relations between the numbers thus, 780= 3.5.52, 153 = 3.51. For convenience of comparison we multiply the numerator and 265 denominator of by 5 ; the two original fractions are then 1351 . 1325 and 15.52 15.51' so that we can put Archimedes' assumption in the form 1351 > W3 ™, 52 ^ 51 and this is seen to be equivalent to 26-^>15V3>26-JI. Now 26 — ^ = /y/262— 1 + i and the latter expression is an approximation to \/262 — 1. We have then 26 - > N/262-l. As 26 —~ was compared with 15is/3, and we want an ap-proximation to J3 itself, we divide by 15 and so obtain iH26^)^^1- But 1 JW=l = y67^1 = = V3, and it follows that U26~^y^- The lower limit for J 3 was given by ^3>A(26-5l)' H. A. / Ixxxii INTRODUCTION. and a glance at this suggests that it may have been arrived at by simply substituting (52 — 1) for 52. Now as a matter of fact the following proposition is true. If c?±b is a whole number which is not a square, while a" is the nearest square number (above or below the first number, as the case may be), then a ±i- > \j'a' ±b > a ± •2a - -2a+l" Hultsch proves this pair of inequalities in a series of propositions formulated after the Greek manner, and there can be little doubt that Archimedes had discovered and proved the same results in substance, if not in the same form. The following circumstances confirm the probability of this assumption. (1) Certain approximations given by Heron show that he knew and frequently used the formula \la2 ±b c\> a ±^- , 'la (where the sign o» denotes "is approximately equal to "). 1 Thus he gives ./75~8 + 14' 1_ 16' 11 16" (2) The formula «/«s + b a + ^ ^ is used by the Arabian Alkarkhl (11th century) who drew from Greek sources (Cantor, p. 719 sq.). It can therefore hardly be accidental that the formula si a +b> a±. ~2a ~ ~2a±l gives us what we want in order to obtain the two Archimedean approximations to >/3, and that in direct connexion with one another*. *" Most of the a priori theories as to the origin of the approximations are open to the serious objection that, as a rule, they give series of approximate values in which the two now in question do not follow consecutively, but are separated by others which do not appear in Archimedes. Hultsch's explanation is much preferable as being free from this objection. But it is fair to say that the actual formula used by Hultsch appears in Hunrath's solution of the puzzle ARITHMETIC IN ARCHIMEDES. lxxxiii We are now in a position to work out the synthesis as follows. From the geometrical representation of \/3 as the perpendicular from an angle of an equilateral triangle on the opposite side we obtain n/22 — 1 = «/3 and, as a first approximation, i Using our formula we can transform this at once into or 2-1. / 1\ 5 Archimedes would then square Í 2 — , or ^, and would obtain 25 27 -g-, which he would compare with 3, or — ; i.e. he would put —g— and would obtain 1(5 + 1)>J3, i.e. ^>JE. To obtain a still nearer approximation, he would proceed in the , /26V 676 _ „ 675 , same manner and compare rjg 1 > or 225 ' w ' or 225 ' w"enoe - 99g — i and therefore that ^26 — =- >/3, , . 1351 ,= that is, i^gQ > v o. The application of the formula would then give the result ^r5(26-5S^i)« /t 1326- 1 265 that is, V3>^ra-, or jgg. The complete result would therefore be 1351 ,r 265 W>V3>T53- (Die Berechnung irrationaler Quadratwurzeln vor der Herrschaft der Decimal-brüche, Kiel, 1884, p. 21; cf. Ueber das Ausziehen der Quadratwurzel bei Griechen und Indern, Hadersleben, 1883), and the same formula is implicitly used in one of the solutions suggested by Tannery (Sur la mesure du cercle d'Archimede in Mémoires de la eociété des sciences physiques et naturelles de Bordeaux, 2" série, it. (1882), p. 313-337). /2 lxxxiv INTRODUCTION. Thus Archimedes probably passed from the first approximation I to |, from | to > anc* ^rom jjj directly to ^fj^ > tne cl°sest approximation of all, from which again he derived the less close 265 approximation . The reason why he did not proceed to a still 1 Do 1351 nearer approximation than ^gQ* is probably that the squaring of this fraction would have brought in numbers much too large to be conveniently used in the rest of his calculations. A similar reason 5 7 will account for his having started from ^ instead of jj if he had used the latter, he would first have obtained, by the same method, J3 =,y/—^ g ^ > and thence - ^ > V3, or |^>v/'3; the squaring of would have given \/'3 = ^gg—~» an<^ *he corresponding approximation would have given gg^^J^ > where again the numbers are inconveniently large for his purpose. § 8. Approximations to the square roots of large numbers. Archimedes gives in the Measurement of a circle the following approximate values: (1) 3013|> 79082321, (2) 1838^ > V3380929, (3) 10091 > VI018405, (4) 2017i> 74069284^, (5) 591i + w2, and dividing 22221 by 2.3010 we see that w=3. Thus 3013 is the integral portion of the square root, and the remainder is 22221 -(2. 3010. 3 + 33), or 4152. The conditions of the proposition now require that the approximate value to be taken for the square root must not be less than lxxxvi INTRODUCTION. the real value, and therefore the fractional part to be added to 3013 must be if anything too great. Now it is easy to see that the 1 1 /iy fraction to be added is greater than ^ because 2 . 3013. ^ + f ^J is less than the remainder 4152. Suppose then that the number required (which is nearer to 3014 than to 3013) is 3014--, and ^ has to be if anything too small. Now (3014)2 = (3013)3 + 2. 3013 + 1 = (3013)* + 6027 = 9082321 -4152 + 6027, whence 9082321 = (3014)2 - 1875. By applying Archimedes' formula J a2 + bV908232T. The required value - has therefore to be not greater than . q " o02o P 1 It remains to be explained why Archimedes put for ^ the value j which is equal to If?! ■ In the first place, he evidently preferred fractions with unity for numerator and some power of 2 for denominator because they contributed to ease in working, e.g. when two such fractions, being equal to each other, had to be added. 9 1 (The exceptions, the fractions and g, are to be explained by exceptional circumstances presently to be mentioned.) Further, in the particular case, it must be remembered that in the subsequent work 2911 had to be added to 3014 — - and the sum divided by 780, q 3 or 2. 2. 3 . 5. 13. It would obviously lead to simplification if a factor could be divided out, e.g. the best for the purpose, 13. Now, dividing 2911 + 3014, or 5925, by 13, we obtain the quotient 455, and a remainder 10, so that 10-- remains to be divided by 13. q J Therefore - has to be so chosen that lOq—p is divisible by 13, while - approximates to, but is not greater than, ^7^5. The solution p = l, q = i would therefore be natural and easy. ARITHMETIC IN ARCHIMEDES. lxxxvii (2) s/3380929. The usual process for extraction of the square root gave as the integral part of it 1838, and as the remainder 2685. As before, it was easy to see that the exact root was nearer to 1839 than to 1838, and that v/3380929 = 1838s + 2685 = 1839s - 2.1838 - 1 + 2685 = 1839*-992. The Archimedean formula then gave 992 1839 - 2-fg39> s/3380929. It could not have escaped Archimedes that i was a near approxima- .. . 992 1984 . 1 1839 , 1 ... .. „ , tion to gg^g or ^ggg, since ^ = ; and ^ would have satisfied the necessary condition that the fraction to be taken must be less 2 than the real value. Thus it is clear that, in taking yj- as the approximate value of the fraction, Archimedes had in view the simplification of the subsequent work by the elimination of a factor. If the fraction be denoted by -, the sum of 1839 — - and 1823, or 3662 -|, had to be divided by 240, i.e. by 6. 40. Division of 3662 by 40 gave 22 as remainder, and then p, q had to be so chosen that P . P 22 — - was conveniently divisible by 40, while - was less than but q J J q 992 approximately equal to ggyg • The solution p = 2, q = 11 was easily seen to satisfy the conditions. (3) Jl 018405. The usual procedure gave 1018405 = 1009s + 324 and the approximation Q04. _ 1009 «7^5 > n/1018405. 324 It was here necessary that the fraction to replace ^Qig should be greater but approximately equal to it, and i satisfied the conditions, while the subsequent work did not require any change in it. lxxxviii INTRODUCTION. (4) ^4069284^- The usual process gave 4069284^ = 2017» + 995^-; it followed that and 2017J was an obvious value to take as an approximation somewhat greater than the left side of the inequality. (5) V349450. In the case of this and the two following roots an approximation had to be obtained which was less, instead of greater, than the true value. Thus Archimedes had to use the second part of the formula b ,-T-r b a±„- > Jar±b>a± a ., . 2a v 2a ±1 In the particular case of 49 450 the integral part of the root is 591, and the remainder is 169. This gave the result and since 169 = 13", while 2.591 + 1=7.13*, it resulted without further calculation that ^349450 > 59If Why then did Archimedes take, instead of this approximation, another which was not so close, viz. 591^? The answer which the subsequent working and the other approximations in the first part of the proof suggest is that he preferred, for convenience of calculation, to use for his approximations fractions of the form i only. But he could not have failed to see that to take the nearest fraction of this form, i, instead of ^ might conceivably affect his final result and 8 7 make it less near the truth than it need be. As a matter of fact, as Hultsch shows, it does not affect the result to take 591^- and to work onwards from that figure. Hence we must suppose that Archimedes had satisfied himself, by taking 591^ and proceeding on that basis for some distance, that he would not be introducing any appreciable error in taking the more convenient though less accurate approximation 591£. ARITHMETIC IN ARCHIMEDES. lxxxix (6) Vl373943ff. In this case the integral portion of the root is 1172, and the remainder 359§§. Thus, if R denote the root, QSOS3 jK>U72 + 2.1172 + 1 359 > 1172 + 2.1172 + Vaf°rti0ri- 359 Now 2.1172 + 1 = 2345; the fraction accordingly becomes -r-r, 1 / 359 \ and = (= 2513 ) sa*isnes the necessary conditions, viz. that it must be approximately equal to, but not greater than, the given fraction. Here again Archimedes would have taken 1172^- as the approximate value but that, for the same reason as in the last case, 1172^- was more convenient. (7) N/5472132rV. The integral portion of the root is here 2339, and the remainder 1211-j-1^, so that, if R is the exact root, ^>2339 + 2JS3TO > 2339J, a fortiori. A few words may be added concerning Archimedes' ultimate reduction of the inequalities 3+ 667*>x>3 + 284i 3 + 4673^>,r>3 + 20T7i to the simpler result 3 ^ > it > 3 ~ . 1 6671 As a matter of fact - = ^qj^, so that in the first fraction it was only necessary to make the small change of diminishing the denominator by 1 in order to obtain the simple 3 j. 2844- 1137 As regards the lower limit for ir, we see that gQp^ = 8069' an<* Hultsch ingeniously suggests the method of trying the effect of increasing the denominator of the latter fraction by 1. This xc INTRODUCTION. 1137 379 produces gQ^Q or gggQ > an(^> ^ we divide 2690 by 379, the quotient is between 7 and 8, so that 1 37_9_ 1 7 > 2690 * 8' Now it is a known proposition (proved in Pappus vn. p. 689) that, it 7- > , then =■ > ^-,. b d b b +d Similarly it may be proved that a + c c b + d d' It follows in the above case that 379 379 + 1 1 2690 2690 + 8 8' which exactly gives ST > 5 > 71 8 10 379 1 and =^ is very much nearer to 7^7; than 3- is. 71 JoyU o Note on alternative hypotheses with regard to the approximations to For a description and examination of all the various theories put forward, up to the year 1882, for the purpose of explaining Archimedes' approximations to V3 the reader is referred to the exhaustive paper by Dr Siegmund Günther, entitled Die quadratischen Irrationalitäten der Alten und deren E-ntwickelungsmethoden (Leipzig, 1882). The same author gives further references in his Abriss der Geschichte der Mathematik und der Naturwissenschaften im, Altertum forming an Appendix to Vol. v. Pt. 1 of Iwan von Müllems Handbuch der klassischen Altertums-vfissenschaft (München, 1894). Günther groups the different hypotheses under three general heads : (1) those which amount to a more or less disguised use of the method of continued fractions and under which are included the solutions of De Lagny, Mollweide, Hauber, Buzengeiger, Zeuthen, P. Tannery (first solution), Heilermann; (2) those which give the approximations in the form of a series of fractions such as a + — + —- H-----(-...; under this class come the solutions of Kadicke, v. Pessl, Rodet (with reference to the (Julvasutras), Tannery (second solution); ARITHMETIC IN ARCHIMEDES. XC1 (3) those which locate the incommensurable surd between a greater and lesser limit and then proceed to draw the limits closer and closer. This class includes the solutions of Oppermann, Alexejeff, Schonborn, Hunrath, though the first two are also connected by Giinther with the method of continued fractions. Of the methods so distinguished by Giinther only those need be here referred to which can, more or less, claim to rest on a historical basis in the sense of representing applications or extensions of principles laid down in the works of Greek mathematicians other than Archimedes which have come down to us. Most of these quasi-historical solutions connect themselves with the system of side- and diagonal-numbers (irXevpiKot and htafierpucoi dpitf/xoi) explained by Theon of Smyrna (c. 130 ad.) in a work which was intended to give so much of the principles of mathematics as was necessary for the study of the works of Plato. The side- and diagonal-numbers are formed as follows. We start with two units, and (a) from the sum of them, (6) from the sum of twice the first unit and once the second, we form two new numbers ; thus 1.1 + 1 = 2, 2.1 + 1 = 3. Of these numbers the first is a side- and the second a diagonal-number respectively, or (as we may say) — 2^ c?2 ~~ 3» In the same way as these numbers were formed from %=1, <^i = l, successive pairs of numbers are formed from a2, d2, and so on, in accordance with the formula a» + i = a» + <4, dn+1 = 2an+dn, whence we have a3=1.2+3 = 5, 2=—g— ARITHMETIC IN ARCHIMEDES. XC111 was well known, Zeuthen concludes that there could have been no difficulty in deducing from Eucl. ii. 5 the identity n / «, /»»S-3»lY /m2 + 3»l2\2 from which, by multiplying up, it was easy to obtain the formula 3 (2mnf + (m2 - 3»2)2 = (m2+3»2)2. If therefore one solution — 3m2 = 1 was known, a second could at once be found by putting x = m?+3n2, y=2mn. Now obviously the equation «i2-3ji2=1 is satisfied by the values m = 2, n=\ ; hence the next solution of the equation tf2-3j/2=l is «1 = 22 + 3.1 = 7, ^ = 2.2.1 = 4; and, proceeding in like manner, we have any number of solutions as ;s2 = 72 + 3 . 42 = 97, 2/2 = 2-7.4=56, x3=972 + 3. 562=18817, j/3= 2.97.66 = 10864, and so on. Next, addressing himself to the other equation ^2-3y2= -2, Zeuthen uses the identity (m + 3n)2 - 3 (m + nf = - 2 {m? - 3»2). Thus, if we know one solution of the equation m2 — 3n2=l, we can proceed to substitute x=m + 3n, y = m + n. Suppose «i = 2, «=1, as before ; we then have ^i = 5> #i = 3-If we put ^2=^ + 3^ = 14, y2=«l+y1 = 8, we obtain x2 = 14 = 7 2/2 8 4 (and m = 7, n=4 is seen to be a solution of to2 —3»i2 = l). Starting again from #2, y2, we have . Xn 19 and -* = — (m = 19, »=11 being a solution of the equation to2 — 3)(2=-2); ^4 = 104, ^=60, *4 26 XC1V INTRODUCTION. (and m=26, n = 15 satisfies m2-3m2 = l), ;r5=284, y5=164, xh 71 Vi 41 „. .. . x6 97 x, 265 , Sim.larly-* = ^, j = ^, and so on. This method gives all the successive approximations to V3, taking account as it does of both the equations «2-3y2 = l, a-2-3y2=-2. Tannery's first solution. Tannery asks himself the question how Diophantus would have set about solving the two indeterminate equations. He takes the first equation in the generalised form xa—ay2 = l, and then, assuming one solution (p, q) of the equation to be known, he supposes p^mx-p, qx=x + q. Then pf- aqfsmPa? - 2mpx +p*—ax*—2aqx -0^ = 1, whence, since j)2—aga-=l, by hypothesis, x=s mp + aq in?-a ' so that l^_+a)p + 2amq amp+^n^ and J>!2 — aq^=l. The values of plt q1 so found are rational but not necessarily integral; if integral solutions are wanted, we have only to put p1 = (w2 + av2)p + 2auvq, qx=2puv + (w2 + av*) q, where (u, v) is another integral solution of x1 - ayi = 1. Generally, if (p, q) be a known solution of the equation x2— ay2=r, suppose py = ap +Pq, qi=yP + &q, and "il suffit pour determiner a,/3,y, 8 de connaitre les trois groupes de solutions les plus simples et de resoudre deux couples d'equations du premier degrt a deux inconnues." Thus (1) for the equation ai2-3y2 = l, the first three solutions are (JJ = 1,2=0), (p=2,q=l), (p = 1,q=4), whence lZ°} and Il^ts}' so that a = 2,/3=3, y = l, 8=2, ARITHMETIC IN ARCHIMEDES. XCV and it follows that the fourth solution is given by p = 2. 7 + 3. 4=26, 2 = 1.7 + 2.4 = 15; (2) for the equation afl — 3y2 = — 2, the first three solutions being (1, 1), (5, 3), (19, 11), we have 5 = a + /3l , 19 = 5a + 3j31 3=y+8j ana 11 = 57+38/' whence a = 2, |3=3, y=l, 8=2, and the next solution is given by _p=2.19 + 3.11 = 71, s = 1.19 + 2.11 = 41, and so on. Therefore, by using the two indeterminate equations and proceeding as shown, all the successive approximations to can be found. Of the two methods of dealing with the equations it will be seen that Tannery's has the advantage, as compared with Zeuthen's, that it can be applied to the solution of any equation of the form xi—ay'i=r. Be Lagny's method. The argument is this. If ^3 could be exactly expressed by an improper fraction, that fraction would fall between 1 and 2, and the square of its numerator would be three times the square of its denominator. Since this is impossible, two numbers have to be sought such that the square of the greater differs as little as possible from 3 times the square of the smaller, though it may be either greater or less. De Lagny then evolved the following successive relations, 22 = 3.12 + 1, 52=3.32-2, 72 = 3 . 42 + l, 192 = 3.112-2, 262 = 3.152+1, 712=3.412-2, etc. From these relations were derived a series of fractions greater than «/3, viz. ^> e*c-i and another series of fractions less than V3, viz. I' IT' 4~T' 6^C' '^'ie ^aW °^ f°rma^on was f°und in each case to be that, if was one fraction in the series and —, the next, then 2 2 p' = 2p + 3q q' p + 2q ' This led to the results 2 7 26 97 362 1351 T>4>l5>56>209> 780 •••>' and 5 19 71 265 989 3691 3 < IT < 41 < 153 < 571 < 2131''" < XCV1 INTRODUCTION. while the law of formation of the successive approximations in each series is precisely that obtained by Tannery as the result of treating the two indeterminate equations by the Diophantine method. ffeilermann's method. This method needs to be mentioned because it also depends upon a generalisation of the system of side- and diagoncd-numbers given by Theon of Smyrna. Theon's rule of formation was S^S^ + B^, Bn=aSn.1 +!)„-,■, and Heilermann simply substitutes for 2 in the second relation any arbitrary number a, developing the following scheme, Sz = S2 + D2, D3=aS2 + D.2, >Sn = >S,t-l + -0n-l> -Dit = «'S'it-l+-DK-l- It follows that By subtraction, Dn* - aSn*= (1 - a) (Dn_* - aSn_*) = (1 -af (D^-aS^), similarly, = (l-a)»W-a^). This corresponds to the most general form of the " Pellian" equation x1 — ay2 = (const.). If now we put B0 = S0= 1, we have D*-„ , (l-«)" + 1 from which it appears that, where the fraction on the right-hand side approaches zero as n increases, ^ is an approximate value for \la. Clearly in the case where a=3, Z>0=2, S0=l we have S^~V «i-3' A'j- 8 ~4' S3~W £4~30~15' A = 71 i>a_194 = 97 Dj = 265 S6~4l' ^ = -26-= 25-' aDd V3~3-102' °r 153' „ _208 102. 27 106 _ 5404 3_ 25 ' 3_ 25.25 + 25 ~25.25' ,- 5404 5 1351 and V3~ 25T208- 3' °rW This is one of the very few instances of success in bringing out the two Archimedean approximations in immediate sequence without any foreign values intervening. No other methods appear to connect the two values in this direct way except those of Hunrath and Hultsch depending on the formula a±^>^aJ±b>a±~—. 2a 2a±l We now pass to the second class of solutions which develops the approximations in the form of the sum of a series of fractions, and under this head comes Tannery's second method. This may be exhibited by means of its application (1) to the case of the square root of a large number, e.g. V349450 or *JblVi + 23409, the first of the kind appearing in Archimedes, (2) to the case of V3- (1) Using the formula Va2 + 6~a+2^, we try the effect of putting for V571J + 23409 the expression 23409 571 + 1142 It turns out that this gives correctly the integral part of the root, and we now suppose the root to be 571 + 20 + i. TO Squaring and regarding ^ as negligible, we have 1142 40 5712+400 + 22840 + ii^f + ™=5 712 + 23409, m m H. A. 9 XCV1U INTRODUCTION. 1182 whence-- = 169, TO 1 169 1 and m=H82>7' 1 7" so that V349450 >59li (2) Bearing in mind that Va2 + ocvJg + ,l ^, . , 2a+l we have V3 = \/12 + 2oj1 + 2.1 + 1 , 2 5 ~l+g, or 3. Assuming then that >JZ = + , squaring and neglecting , we obtain 9 ^3m ' whence m = 15, and we get as the second approximation 5,1 26 3+15' °r IS" We have now 262 - 3.152 = 1, and can proceed to find other approximations by means of Tannery's first method. / 2 1 1\2 Or we can also put (1 +- + + - I =3, and, neglecting , we get 262 62 whence n= —15. 52 = - 780, and /s /, 2 1 1 1351\ V3~(1 + 3 + l5-780 = -780> 1351 It is however to be observed that this method only connects with 7oU and not with the intermediate approximation —r, to obtain which 15 loo Tannery implicitly uses a particular case of the formula of Hunrath and Hultsch. Rodet's method was apparently invented to explain the approximation in the Qulvasiitras* ^200!+-+ —• 3 3.4 3.4.34' * See Cantor, Vorlesungen über Gesch. d. Math. p. 600 sq. ARITHMETIC IN ARCHIMEDES. XC1X 4 but, given the approximation -, the other two successive approximations 6 indicated by the formula can be obtained by the method of squaring just described* without such elaborate work as that of Rodet, which, when applied to sjz, only gives the same results as the simpler method. Lastly, with reference to the third class of solutions, it may be mentioned (1) that Oppermann used the formula a + b 2ab -2->V«6>^-6> 2 /- 3 which gave successively ^ > V 3 > ^, but only led to one of the Archimedean approximations, and that by combining the last two ratios, thus 97 + 168 265 56 + 97 ~153' (2) that Schonborn came somewhat near to the formula successfully used by Hunrath and Hultsch when he proved+ that a + ^->^/a2 + b>a-\--^—=. ~2a ~ - 2a±V6 * Cantor had already pointed this out in his first edition of 1880. + Zeitschrift fur Math. u. Physik (Hist. litt. Abtheilung) xxvin. (1883), p. 169 sq. 92