Notes

¹By group we mean finite group, that is, one which cannot be brought into one-to-one correspondence (§ 2) with any part of itself.

²A definition in terms of symbolic, not numerical addition. The sign + can, of course, indicate numerical addition only when both the symbols which it connects are numbers.

³On the other hand, $-a$ is said to be numerically greater than, equal to, or less than $-b$, according as $a$ is itself greater than, equal to, or less than $b$.

⁴In this connection see § 25.

⁵The case $b=0$ is excluded, 0 not being a number in the sense in which that word is here used.

⁶The doctrine of symbolic division admits of being presented in the very same form as that of symbolic subtraction. The equations of Chapter II immediately pass over into theorems respecting division when the signs of multiplication and division are substituted for those of addition and subtraction; so, for instance,

In particular, to corresponds $\frac{a}{a}a=a$. Thus a purely symbolic definition may be given 1. It plays the same rôle in multiplication as 0 in addition. Again, it has the same exceptional character in involution—an operation related to multiplication quite as multiplication to addition—as 0 in multiplication; for $1^m=1^n$, whatever the values of $m$ and $n$. Similarly, to the equation , or , corresponds , which answers as a definition of the unit fraction $\frac{1}{a}$; and in terms of these unit fractions and integers all other fractions may be expressed.

⁷In this connection see § 32.

⁸These remarks, mutatis mutandis, apply with equal force to subtraction.

⁹It hardly need be said that the fraction, zero, and the negative actually made their way into the number-system for quite a different reason from this;—because they admitted of certain “real” interpretations, the fraction in measurements of lines, the negative in debit where the corresponding positive meant credit or in a length measured to the left where the corresponding positive meant a length measured to the right. Such interpretations, or correspondences to existing things which lie entirely outside of pure arithmetic, are ignored in the present discussion as being irrelevant to a pure arithmetical doctrine of the artificial forms of number.

¹⁰The word “algebra” is here used in the general sense, the sense in which quaternions and the Ausdehungslehre (see §§ 127, 128) are algebras. Inasmuch as elementary arithmetic, as actually constituted, accepts the fraction, there is no essential difference between it and elementary algebra with respect to the kinds of number with which it deals; algebra merely goes further in the use of artificial numbers. The elementary algebra differs from arithmetic in employing literal symbols for numbers, but chiefly in making the equation an object of investigation.

¹¹It is worth noticing that the determinateness of division is here not an independent assumption, but a consequence of the definition of multiplication and the determinateness of the division of rationals. The same thing is true of the other fundamental laws I–V, VII.

¹²What the above demonstration proves is that $a$ stands in the same relation to ${\alpha }_{\mu }$ when irrational as when rational. The principle of permanence (cf. § 12), therefore, justifies one in regarding $a$ as the ideal limit in the former case since it is the actual limit in the latter (§ 27). $a$, when irrational, is limit in precisely the same sense that $\frac{c}{d}$ is the quotient of $c$ by $d$, when $c$ is a positive integer not containing $d$. It follows from the demonstration that if there be a reality corresponding to $a$, as in geometry we assume there is (§ 40), that reality will be the actual limit of the reality of the same kind corresponding to ${\alpha }_{\mu }$. The notion of irrational limiting values was not immediately available because, prior to §§ 28, 29, 30, the meaning of difference and greater and lesser inequality had not been determined for numbers defined by sequences.

¹³Gauss introduced the use of $i$ to represent $\sqrt{-1}$.

¹⁴Throughout this discussion $\infty$ is not regarded as belonging to the number-system, but as a limit of the system, lying without it, a symbol for something greater than any number of the system.

¹⁵What is here proven is that in the system of complex numbers formed from the fundamental units 1 and $i$ there is one, and but one, number which is the quotient of $a+ib$ by $a^{\prime }+i{b}^{\prime }$; this being a consequence of the determinateness of the division of real numbers and the peculiar relation ($i^2=-1$) holding between the fundamental units. For the sake of the permanence of IX we make the assumption, otherwise irrelevant, that this is the only value of the quotient whether within or without the system formed from the units 1 and $i$.

¹⁶That is, in terms of the first powers of these units.

¹⁷Zur Theorie der aus $n$ Haupteinheiten gebildeten complexen Grössen. Göttinger Nachrichten Nr. 10, 1884. Weierstrass finds that these general complex numbers differ in only one important respect from the complex number $a+ib$. If the number of fundamental units be greater than 2, there always exist numbers, different from 0, the product of which by certain other numbers is 0. Weierstrass calls them divisors of 0. The number of exceptions to the determinateness of division is infinite instead of one.

¹⁸These units are, generally speaking, not $e_1,e_2,\dots ,e_n$, but linear combinations of them, as ${\gamma }_1{e}_1+{\gamma }_2{e}_2+\cdots +{\gamma }_n{e}_n$, ${\kappa }_1{e}_1+{\kappa }_2{e}_2+\cdots +{\kappa }_n{e}_n$. Any set of $n$ independent linear combinations of the units $e_1,e_2,\dots ,e_n$ may be regarded as constituting a set of fundamental units, since all numbers of the form ${\alpha }_1{e}_1+{\alpha }_2{e}_2+\cdots +{\alpha }_n{e}_n$ may be expressed linearly in terms of them.

¹⁹A reality has thus been found to correspond to the hitherto uninterpreted symbol $a+ib$. But this reality has no connection with the reality which gave rise to arithmetic, the number of things in a group of distinct things, and does not at all lessen the purely symbolic character of $a+ib$ when regarded from the standpoint of that reality, the standpoint which must be taken in a purely arithmetical study of the origin and nature of the number concept. The connection between the numbers $a+ib$ and the points of a plane is purely artificial. The tangible geometrical pictures of the relations among complex numbers to which it leads are nevertheless a valuable aid in the study of these relations.

²⁰For the demonstration of these, the so-called addition theorems of trigonometry, see Wells’ Trigonometry, § 65, or any other text-book of trigonometry.

²¹This use of the symbol $e^{i𝜃}$ will be fully justified in § 73.

²²Dr. F. Franklin, American Journal of Mathematics, Vol. VII, p. 376. Also Möbius, Collected Works, Vol. IV, p. 726.

²³ In the series $A\rho +B{\rho }^2+C{\rho }^3+$ etc., the ratio of all the terms following the first to the first, i. e.

which by taking $\rho$ small enough may evidently be made as small as we please.

²⁴In the above demonstration it is assumed that the coefficient of $\delta$ is not 0. If it be 0, let $A{\delta }^r$ denote the first term of $\Delta$ which is not 0. If , we then have

from which the same conclusion follows as above.

²⁵An application of the principle of permanence (§12) is involved in the use of functional equations to define functions. The equation $a^z{a}^t=a^{z+t}$, for instance, only becomes a functional equation when its permanence is assumed for other values of $z$ and $t$ than those for which it has been actually demonstrated. In this respect the methods of definition of the negative and the fraction on the one hand, and the functions $a^z$, ${log}_{a}z$, on the other, are identical; but, while the equation itself served as definition of $a-b$, there being no simpler symbols in terms of which $a-b$ could be expressed, from the equation $a^z{a}^t=a^{z+t}$ a series (§ 73, (4)) may be deduced which defines $a^z$ in terms of numbers of the system $a+ib$.

²⁶This number $e$, the base of the Naperian system of logarithms, is a “transcendental” irrational, transcendental in the sense that there is no algebraic equation with integral coefficients of which it can be a root (see Hermite, Comptes Rendus, LXXVII). $\pi$ has the same character, as Lindemann proved in 1882, deducing at the same time the first actual demonstration of the impossibility of the famous old problem of squaring the circle by aid of the straight edge and compasses only (see Mathematische Annalen, XX).

²⁷For instance is not equal to ${log}_{e}z+{log}_{e}t$ for arbitrarily chosen values of these logarithms, but to ${log}_{e}z+{log}_{e}t\pm i2n\pi$, where $n$ is some positive integer.

²⁸$\frac{a^z}{a^t}=a^{z-t}$, which is sometimes included among the fundamental laws to which $a^z$ is subject, follows immediately from $a^z{a}^t=a^{z+t}$ by the definition of division.

²⁹Pure quinary and vigesimal systems are rare, if indeed they occur at all. As an example of the former, Tylor (Primitive Culture, I, p. 261) instances a Polynesian number series which runs 1, 2, 3, 4, 5, $5\cdot 1$, $5\cdot 2$,…; and as an example of the latter, Cantor (Geschichte der Mathematik, p. 8), following Pott, cites the notation of the Mayas of Yucatan who have special words for 20, 400, 8000, 160,000. The Hebrew notation, like the Indo-Arabic, affords an example of a pure decimal notation. Mixed systems are common. Thus the Roman is mixed decimal and quinary, the Aztec mixed vigesimal and quinary. Speaking generally, the quinary and vigesimal systems are more frequent among the lower races, the decimal among the higher. (Primitive Culture, I, p. 262.)

³⁰So, for instance, the aborigines of Victoria and the Bororos of Brazil (Primitive Culture, I, p. 244).

³¹In the language of the Tamanacs on the Orinoco the word for 5 means “a whole hand,” the word for 6, “one of the other hand,” and so on up to 9; the word for 10 means “both hands,” 11, “one to the foot,” and so on up to 14; 15 is “a whole foot,” 16, “one to the other foot,” and so on up to 19; 20 is “one Indian,” 40, “two Indians,” etc. Other languages rich in digit numerals are the Cayriri, Tupi, Abipone, and Carib of South America; the Eskimo, Aztec, and Zulu (Primitive Culture, I, p. 247). “Two” in Chinese is a word meaning “ears,” in Thibet “wing,” in Hottentot “hand.” (Gow, Short History of Greek Mathematics, p. 7.) See also Primitive Culture, I, pp. 252–259.

³²Were there any reason for supposing that primitive counting was done with numeral words, it would be probable that the ordinals, not the cardinals, were the earliest numerals. For the normal order of the cardinals must have been fully recognized before they could be used in counting. In this connection, see Kronecker, Ueber den Zahlbegriff; Journal für die reine und angewandte Mathematik, Vol. 101, p. 337. Kronecker goes so far as to declare that he finds in the ordinal numbers the natural point of departure for the development of the number concept.

³³Dr. Isaac Taylor, in his book “The Alphabet,” names this alphabet the Indo-Bactrian. Its earliest and most important monument is the version of the edicts of King Asoka at Kapur-di-giri. In this inscription, it may be added, numerals are denoted by strokes, as $|,||,|\left|\right|,\left|\right|\left|\right|,\left|\right|\left|\right||$.

³⁴Columns 1–5, 7, 8 of the table on page 89 are taken from Taylor’s Alphabet, II, p. 266; column 6, from Cantor’s Geschichte der Mathematik.

³⁵The Rhind papyrus of the British Museum; translated by A. Eisenlohr, Leipzig, 1877.

³⁶The usual method of expressing fractions was to write the numerator with an accent, and after it the denominator twice with a double accent: e. g. $\iota {\zeta }^{\prime }\kappa {\alpha }^{\prime \prime }\kappa {\alpha }^{\prime \prime }=\frac{17}{21}$. Before sexagesimal fractions came into vogue actual reckonings with fractions were effected by unit fractions, of which only the denominators (doubly accented) were written.

³⁷This is the explicit declaration of the most reliable document extant on the history of geometry before Euclid, a chronicle of the ancient geometers which Proclus (a. d.  450) gives in his commentary on Euclid, deriving it from a history written by Eudemus about 330 b. c. This chronicle credits the Egyptians with the discovery of geometry and Thales (600 b. c. ) with having first introduced this study into Greece. Thales and Pythagoras are the founders of the Greek mathematics. But while Thales should doubtless be credited with the first conception of an abstract deductive geometry in contradistinction to the practical empirical geometry of Egypt, the glory of realizing this conception belongs chiefly to Pythagoras and his disciples in the Greek cities of Italy (Magna Græcia); for they established the principal theorems respecting rectilineal figures. To the Pythagoreans the discovery of many of the elementary properties of numbers is due, as well as the geometric form which characterized the Greek theory of numbers throughout its history. In the middle of the fifth century before Christ Athens became the principal centre of mathematical activity. There Hippocrates of Chios (430 b. c. ) made his contributions to the geometry of the circle, Plato (380 b. c.) to geometric method, Theætetus (380 b. c.) to the doctrine of incommensurable magnitudes, and Eudoxus (360 b. c.) to the theory of proportion. There also was begun the study of the conics. About 300 b. c.  the mathematical centre of the Greeks shifted to Alexandria, where it remained. The third century before Christ is the most brilliant period in Greek mathematics. At its beginning—in Alexandria—Euclid lived and taught and wrote his Elements, collecting, systematizing, and perfecting the work of his predecessors. Later (about 250) Archimedes of Syracuse flourished, the greatest mathematician of antiquity and founder of the science of mechanics; and later still (about 230) Apollonius of Perga, “the great geometer,” whose Conics marks the culmination of Greek geometry. Of the later Greek mathematicians, besides Hero and Diophantus, of whom an account is given in the text, and the great summarizer of the ancient mathematics, Pappus (300 a. d.), only the famous astronomers Hipparchus (130 b. c.) and Ptolemy (150 a. d. ) call for mention here. To them belongs the invention of trigonometry and the first trigonometric tables, tables of chords. The dates in this summary are from Gow’s Hist. of Greek Math.

³⁸Compare Cantor, Geschichte der Mathematik, p. 153.

³⁹His demonstration may easily have been the following, which was old enough in Aristotle’s time (340 b. c.) to be made the subject of a popular reference, and which is to be found at the end of the 10th book in all old editions of Euclid’s Elements: If there be any line which the side and diagonal of a square both contain an exact number of times, let their lengths in terms of this line be $a$ and $b$ respectively; then $b^2=2a^2$. The numbers $a$ and $b$ may have a common factor, $\gamma$; so that $a=\alpha \gamma$ and $b=\beta \gamma$, where $\alpha$ and $\beta$ are prime to each other. The equation $b^2=2a^2$ then reduces, on the removal of the factor ${\gamma }^2$ common to both its members, to ${\beta }^2=2{\alpha }^2$. From this equation it follows that ${\beta }^2$, and therefore $\beta$, is an even number, and hence that $\alpha$ which is prime to $\beta$ is odd. But set $\beta =2{\beta }^{\prime }$, where ${\beta }^{\prime }$ is integral, in the equation ${\beta }^2=2{\alpha }^2$; it becomes $4{\beta }^{\prime 2}=2{\alpha }^2$, or $2{\beta }^{\prime 2}={\alpha }^2$, whence ${\alpha }^2$, and therefore $\alpha$, is even. $\alpha$ has thus been proven to be both odd and even, and is therefore not a number.

⁴⁰The formula for the area of a triangle in terms of its sides is due to Hero.

⁴¹Many attempts have been made to discover the methods of approximation used by Archimedes and Hero from an examination of their results, but with little success. The formula $\sqrt{a^2\pm b}=a\pm \frac{b}{2a}$ will account for some of the simpler approximations, but no single method or set of methods have been found which will account for the more difficult. See Günther: Die quadratischen Irrationalitäten der Alten und deren Entwicklungsmethoden. Leipzig, 1882. Also in Handbuch der klassischen Altertums-Wissenschaft, 11ter. Halbband.

⁴²His symbol for the unknown quantity is the word hau, meaning heap.

⁴³Elements, VI, 29, 28; Data, 84, 85.

⁴⁴Die Lehre von den Kegelschnitten im Altertum. Copenhagen, 1886.

⁴⁵See Cantor; Geschichte der Mathematik, p. 341.

⁴⁶The designation “Diophantine equations,” commonly applied to indeterminate equations of the first degree when investigated for integral solutions, is a striking misnomer. Diophantus nowhere considers such equations, and, on the other hand, allows fractional solutions of indeterminate equations of the second degree.

⁴⁷The mathematical chapters of Brahmagupta and Bhâskara have been translated into English by Colebrooke: “Algebra, Arithmetic, and Mensuration, from the Sanscrit of Brahmagupta and Bhâskara,” 1817; those of Âryabhaṭṭa into French by L. Rodet (Journal Asiatique, 1879).

⁴⁸This method still goes under the name “Hindoo method.”

⁴⁹It was Almanṣûr who transferred the throne of the caliphs from Damascus to Bagdad which immediately became not only the capital city of Islam, but its commercial and intellectual centre.

⁵⁰This translation remained the guide of the Arabian astronomers until the reign of Almamûn (813–833), for whom Alkhwarizmî prepared his famous astronomical tables (820). Even these were based chiefly on the “Sindhind,” though some of the determinations were made by methods of the Persians and Ptolemy.

⁵¹Thus suppose the equation $x^3+bx=a$, given. For $b$ substitute the quantity $p^2$, and for $a$, $p^{2}r$. Then . Now this equation is the result of eliminating $y$ from between the two equations, $x^2=py$, ; the first of which is the equation of a parabola, the second, of a circle. Let these two curves be constructed; they will intersect in one real point distinct from the origin, and the abscissa of this point is a root of $x^3+bx=a$. See Hankel, Geschichte der Mathematik, p. 279. This method is of greater interest in the history of geometry than in that of algebra. It involves an anticipation of some of the most important ideas of Descartes’ Géométrie (see p. 118).

⁵²See §88.

⁵³Besides Leonardo there flourished in the first quarter of the 13th century an able German mathematician, Jordanus Nemorarius. He was the author of a treatise entitled De numeris datis, in which known quantities are for the first time represented by letters, and of one De trangulis which is a rich though rather systemless collection of theorems and problems principally of Greek and Arabian origin. See Günther: Geschichte des mathemathischen Unterrichts im deutschen Mittelalter, p. 156.

⁵⁴Compare Hankel, Geschichte der Mathematik, pp. 349–352. To the unfruitfulness of these centuries the Summa of Luca Pacioli bears witness. This book, which has the distinction of being the earliest book on algebra printed, appeared in 1494, and embodies the arithmetic, algebra, and geometry of the time just preceding the Renaissance. It contains not an idea or method not already presented by Leonardo. Even in respect to algebraic symbolism it surpasses the Liber Abaci only to the extent of using abbreviations for a few frequently recurring words, as p. for “plus,” and R. for “res” (the unknown quantity). And this is not to be regarded as original with Pacioli for the Arabians of Leonardo’s time made a similar use of abbreviations. In a translation made by Gerhard of Cremona (12th century) from an unknown Arabic original the letters r (radix), $c$ (census), $d$ (dragma) are used to represent the unknown quantity, its square, and the absolute term respectively. The Summa of Pacioli has great merits, notwithstanding its lack of originality. It satisfied the mathematical needs of the time. It is very comprehensive, containing full and excellent instruction in the art of reckoning after the methods of Leonardo, for the merchant-man, and a great variety of matter of a purely theoretical interest also—representing the elementary theory of numbers, algebra, geometry, and the application of algebra to geometry. Compare Cantor, Geschichte der Mathematik, II, p. 308. It should be added that the 15th century produced a mathematician who deserves a distinguished place in the general history of mathematics on account of his contributions to trigonometry, the astronomer Regiomontanus (1436–1476). Like Jordanus, he was a German.

⁵⁵The proper title of this work is: “Artis magnae sive de regulis Algebraicis liber unus.” It has stolen the title of Cardan’s “Ars magna Arithmeticae,” published at Basel, 1570.

⁵⁶Mémoire sur les Equations Algébriques: Christiania, 1826. Also in Crelle’s Journal, I, p. 65.

⁵⁷L’Algebra, 1579. He also formally states rules for reckoning with $\pm \sqrt{-1}$ and $a+b\sqrt{-1}$.

⁵⁸There are isolated instances of this use of letters much earlier than Vieta in the De numeris datis of Jordanus Nemorarius, and in the Algorithmus demonstratus of the same author. But the credit of making it the general practice of algebraists belongs to Vieta.

⁵⁹One has only to reflect how much of the power of algebra is due to its admirable symbolism to appreciate the importance of the Artis Analyticae Praxis, in which this symbolism is finally established. But one addition of consequence has since been made to it, integral and fractional exponents introduced by Descartes (1637) and Wallis (1659). Harriot substituted small letters for the capitals used by Vieta, but followed Vieta in representing known quantities by consonants and unknown by vowels. The present convention of representing known quantities by the earlier letters of the alphabet, unknown by the later, is due to Descartes. Vieta’s notation is unwieldy and ill adapted to purposes of algebraic reckoning. Instead of restricting itself, as Harriot’s does, to the use of brief and easily apprehended conventional symbols, it also employs words subject to the rules of syntax. Thus for $A^3-3B^{2}A=Z$ (or $aaa-3bba=z$, as Harriot would have written it), Vieta writes A cubus - B quad 3 in A aequatur Z solido. In this respect Vieta is inferior not only to Harriot, but to several of his predecessors and notably to his contemporary, the Dutch mathematician Stevinus (1548–1620), who would, for instance, have written $x^2+3x-8$ as $1\ast +3\ast -8\ast$. The geometric affiliations of Vieta’s notation are obvious. It suggests the Greek arithmetic. It is surprising that algebraic symbolism should owe so little to the great Italian algebraists of the 16th century. Like Pacioli (see note, p. 113) they were content with a few abbreviations for words, a “syncopated” notation, as it has been called, and an incomplete one at that. The current symbols of operation and relation are chiefly of English and German origin, having been invented or adopted as follows: viz. $=$, by Recorde in 1556; $\sqrt{}$, by Rudolf in 1525; the vinculum, by Vieta in 1591; brackets, by Bombelli, 1572; $÷$, by Rahn in 1659; $\times ,>,<,$ by Harriot in 1631. The signs $+$ and $-$ occur in a 15th century manuscript discovered by Gerhardt at Vienna. The notations $a-b$ and $\frac{a}{b}$ for the fraction were adopted from the Arabians.

⁶⁰See Géométrie, Livre II. In Cousin’s edition of Descartes’ works, Vol. V, p. 337.

⁶¹Descartes fails to recognize a number of the conventions of our modern Cartesian geometry. He makes no formal choice of two axes of reference, calls abscissas $y$ and ordinates $x$, and as frequently regards as positive ordinates below the axis of abscissas as ordinates above it.

⁶²Géométrie, Livre I. Ibid. pp. 313–314.

⁶³Géométrie, Livre II. Ibid. pp. 337–338.

⁶⁴See W. W. Beman in Proceedings of the American Association for the Advancement of Science, 1897.

⁶⁵See Gauss, Complete Works, II, p. 174.

⁶⁶Arithmetical and Symbolical Algebra, 1830 and 1845; especially the later edition. Also British Association Reports, 1833.

⁶⁷Algebra, edition of 1845, §§ 631, 569, 639.

⁶⁸Algebra, Appendix, §631.

⁶⁹Ibid. §557.

⁷⁰Gergonne’s Annales, 1813. One must go back to Euclid for the earliest known recognition of any of these laws. Euclid demonstrated, of integers (Elements, VII, 16), that $ab=ba$.

⁷¹In 1838. See The Mathematical Writings of D. F. Gregory, p. 2. Among other writings of this period, which promoted a correct understanding of the artificial numbers, should be mentioned Gregory’s interesting paper, “On a Difficulty in the Theory of Algebra,” Writings, p. 235, and De Morgan’s papers “On the Foundation of Algebra” (1839, 1841; Cambridge Philosophical Transactions, VII).

⁷²Philosophical Magazine, II, Vol. 25, 1844.

⁷³See Cantor in Mathematische Annalen, V, p. 123, XXI, p. 567. The first paper was written in 1871. In the second, Cantor compares his theory with that of Weierstrass, and also with the theory proposed by Dedekind in his Stetigkeit und irrationals Zahlen (1872). The theory of the irrational, set forth in Chapter IV of the first part of this book, is Cantor’s.

⁷⁴Journal für die reine und angewandte Mathematik, Vol. 101, p. 337.

⁷⁵Göttinger Nachrichten for 1884, p. 395; 1885, p. 141; 1889, p. 34, p. 237. Leipziger Berichte for 1889, p. 177, p. 290, p. 400. Mathemathische Annalen, XXXIII, p. 49.

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