The above facts and formulae admit of being stated in a great variety of ways according to the notation and the particular letters used. Consequently the summaries which have been given of Eucl. x. by various writers differ much in appearance while expressing the same thing in substance. The first summary in algebraical form (and a very elaborate one) seems to have been that of Cossali (Origine, trasporto in Italia, primi progressi in essa dell' Algebra, Vol. 11. pp. 242-65) who takes credit accordingly (p. 265). In 1794 Meier Hirsch published at Berlin an Algebraischer Commentar über das zehente Buch der Elemente des Euklides which gives the contents in algebraical form but fails to give any indication of Euclid's methods, using modern forms of proof only. In 1834 Poselger wrote a paper, Ueber das zehnte Buch der Elemente des Euklides, in which he pointed out the defects of Hirsch's reproduction and gave a summary of his own, which however, though nearer to Euclid's form, is difficult to follow in consequence of an elaborate system of abbreviations, and is open to the objection that it is not algebraical enough to enable the character of Euclid's irrationals to be seen at a glance. Other summaries will be found (1) in Nesselmann, Die Algebra der Griechen, pp. 165-84; (2) in Loria, Il periodo aureo della geometria greca, Modena, 1895, pp. 40-9; (3) in Christensen's article "Ueber Gleichungen vierten Grades im zehnten Buch der Elemente Euklids" in the Zeitschrift für Math. u. Physik (Historisch-literarische Abtheilung), XXXIV. (1889), pp. 201-17. The only summary in English that I know is that in the Penny Cyclopaedia, under "Irrational quantity," by De Morgan, who yielded to none in his admiration of Book x. "Euclid investigates," says De Morgan, "every possible variety of lines which can be represented by √(√a ± √b), a and b representing two commensurable lines.... This book has a completeness which none of the others (not even the fifth) can boast of: and we could almost suspect that Euclid, having arranged his materials in his own mind, and having completely elaborated the 10th Book, wrote the preceding books after it and did not live to revise them thoroughly."

Thus

Much attention was given to Book x. by the early algebraists. Leonardo of Pisa (fl. about 1200 A.D.) wrote in the 14th section of his Liber Abaci on the theory of irrationalities (de tractatu binomiorum et recisorum), without however (except in treating of irrational trinomials and cubic irrationalities) adding much to the substance of Book x.; and, in investigating the equation

x+2x2+10x= €20,

propounded by Johannes of Palermo, he proved that none of the irrationals in Eucl. x. would satisfy it (Hankel, pp. 344-6, Cantor, 111, p. 43). Luca Paciuolo (about 1445-1514 A.D.) in his algebra based himself largely, as he himself expressly says, on Euclid x. (Cantor, II, p. 293). Michael Stifel (1486 or 1487 to 1567) wrote on irrational numbers in the second Book of his Arithmetica integra, which Book may be regarded, says Cantor (11,, p. 402), as an elucidation of Eucl. x. The works of Cardano (1501-76) abound in speculations regarding the irrationals of Euclid, as may be seen by reference to Cossali (Vol. II., especially pp. 268-78 and 382-99); the character of the various odd and even powers of the binomials and apotomes is therein investigated, and Cardano considers in detail of what particular forms of equations, quadratic, cubic, and biquadratic, each class of Euclidean irrationals can be roots. Simon Stevin (1548-1620) wrote a Traité des incommensurables grandeurs en laquelle est sommairement déclaré le contenu du Dixiesme Livre d'Euclide (Oeuvres mathématiques, Leyde, 1634, pp. 219 sqq.); he speaks thus

INTRODUCTORY NOTE

9

of the book: "La difficulté du dixiesme Livre d'Euclide est à plusieurs devenue en horreur, voire jusque à l'appeler la croix des mathématiciens, matière trop dure à digérer, et en la quelle n'aperçoivent aucune utilité," a passage quoted by Loria (Il periodo aureo della geometria greca, p. 41).

It will naturally be asked, what use did the Greek geometers actually make of the theory of irrationals developed at such length in Book x.? The answer is that Euclid himself, in Book XIII., makes considerable use of the second portion of Book x. dealing with the irrationals affected with a negative sign, the apotomes etc. One object of Book XIII. is to investigate the relation of the sides of a pentagon inscribed in a circle and of an icosahedron and dodecahedron inscribed in a sphere to the diameter of the circle or sphere respectively, supposed rational. The connexion with the regular pentagon of a straight line cut in extreme and mean ratio is well known, and Euclid first proves (XIII. 6) that, if a rational straight line is so divided, the parts are the irrationals called apotomes, the lesser part being a first apotome. Then, on the assumption that the diameters of a circle and sphere respectively are rational, he proves (XIII. II) that the side of the inscribed regular pentagon is the irrational straight line called minor, as is also the side of the inscribed icosahedron (XIII. 16), while the side of the inscribed dodecahedron is the irrational called an apotome (XIII. 17).

Of course the investigation in Book x. would not have been complete if it had dealt only with the irrationals affected with a negative sign. Those affected with the positive sign, the binomials etc., had also to be discussed, and we find both portions of Book x., with its nomenclature, made use of by Pappus in two propositions, of which it may be of interest to give the enunciations here.

If, says Pappus (IV. p. 178), AB be the rational diameter of a semicircle, and if AB be produced to C so that BC is equal to the radius, if CD be a tangent,

if E be the middle point of the arc BD, and if CE be joined, then CE is the irrational straight line called minor. As a matter of fact, if p is the radius,

CE2= p2 (5-2√3) and CE =

√5

5+ √135-√13

2

2

If, again (p. 182), CD be equal to the radius of a semicircle supposed

B

K

F

rational, and if the tangent DB be drawn and the angle ADB be bisected by DF meeting the circumference in F, then DF is the excess by which the binomial exceeds the straight line which produces with a rational area a medial

ΙΟ

whole (see Eucl. x. 77). irrational straight line.)

(In the figure DK is the binomial and KF the other As a matter of fact, if p be the radius,

, and KF p. √√3-1=P.

ρ.

Proclus tells us that Euclid left out, as alien to a selection of elements, the discussion of the more complicated irrationals, "the unordered irrationals which Apollonius worked out more fully" (Proclus, p. 74, 23), while the scholiast to Book x. remarks that Euclid does not deal with all rationals and irrationals but only the simplest kinds by the combination of which an infinite number of irrationals are obtained, of which Apollonius also gave some. The author of the commentary on Book x. found by Woepcke in an Arabic translation, and above alluded to, also says that "it was Apollonius who, beside the ordered irrational magnitudes, showed the existence of the unordered and by accurate methods set forth a great number of them." It can only be vaguely gathered, from such hints as the commentator proceeds to give, what the character of the extension of the subject given by Apollonius may have been. See note at end of Book.

DEFINITIONS.

I. Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.

2. Straight lines are commensurable in square when the squares on them are measured by the same area, and incommensurable in square when the squares on them cannot possibly have any area as a common measure.

3. With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. Let then the assigned straight line be called rational, and those straight lines which are commensurable with it, whether in length and in square or in square only, rational, but those which are incommensurable with it irrational.

4. And let the square on the assigned straight line be called rational and those areas which are commensurable with it rational, but those which are incommensurable with it irrational, and the straight lines which produce them irrational, that is, in case the areas are squares, the sides themselves, but in case they are any other rectilineal figures, the straight lines on which are described squares equal to them.

Εὐθεῖαι δυνάμει σύμμετροί εἰσιν, ὅταν τὰ ἀπ ̓ αὐτῶν τετράγωνα τῷ οἱ μετρῆται, ἀσύμμετροι δέ, ὅταν τοῖς ἀπ ̓ αὐτῶν τετραγώνοις μηδὲν ἐνδέχη κοινὸν μέτρον γενέσθαι.

Commensurable in square is in the Greek δυνάμει σύμμετρος. translations (e.g. Williamson's) dvráμe has been translated "in po as the particular power represented by duvaus in Greek geometry I have thought it best to use the latter word throughout. It will be that Euclid's expression commensurable in square only (used in D constantly) corresponds to what Plato makes Theaetetus call a s (dúvapus) in the sense of a surd. If a is any straight line, a and am and an (where m, n are integers or arithmetical fractions lowest terms, proper or improper, but not square) are commensurable only. Of course (as explained in the Porism to x. 10) all stra commensurable in length (pýκei), in Euclid's phrase, are commens square also; but not all straight lines which are commensurable in commensurable in length as well. On the other hand, straight lin mensurable in square are necessarily incommensurable in length also all straight lines which are incommensurable in length are incomm in square. In fact, straight lines which are commensurable in squar incommensurable in length, but obviously not incommensurable in s

DEFINITION 3.

Τούτων ὑποκειμένων δείκνυται, ὅτι τῇ προτεθείσῃ εὐθείᾳ ὑπάρχουσ πλήθει ἄπειροι σύμμετροί τε καὶ ἀσύμμετροι αἱ μὲν μήκει μόνον, αἱ δὲ κα καλείσθω οὖν ἡ μὲν προτεθεῖσα εὐθεῖα ῥητή, καὶ αἱ ταύτῃ σύμμετροι εἴτε δυνάμει εἴτε δυνάμει μόνον ῥηταί, αἱ δὲ ταύτῃ ἀσύμμετροι ἄλογοι καλείσθ

The first sentence of the definition is decidedly elliptical. I strictly speaking, assert that "with a given straight line there are a number of straight lines which are (1) commensurable either (a) only or (b) in square and in length also, and (2) incommensurab (a) in length only or (b) in length and in square also."

The relativity of the terms rational and irrational is well broug this definition. We may set out any straight line and call it ration is then with reference to this assumed rational straight line that c called rational or irrational.

We should carefully note that the signification of rational in Eucli than in our terminology. With him, not only is a straight line commen length with a rational straight line rational, but a straight line is ratio is commensurable with a rational straight line in square only. That i p rational, where m, n are int

rational straight line, not only is

m

n

m

n

m

in this case call p irrational. It would appear that Euclid's terminology here differed as much from that of his predecessors as it does from ours. We are familiar with the phrase άρρητος διάμετρος τῆς πεμπάδος by which Plato (evidently after the Pythagoreans) describes the diagonal of a square on a straight line containing 5 units of length. This "inexpressible diameter of five (squared)" means √50, in contrast to the pnrn diáμerpos, the "expressible diameter" of the same square, by which is meant the approximation √50-1, or 7. Thus for Euclid's predecessors would apparently not have been rational but appηros, "inexpressible," i.e. irrational. I shall throughout my notes on this Book denote a rational straight line in Euclid's sense by p, and by p and σ when two different rational straight lines are required. Wherever then I use p or σ, it must be remembered that p, σ may have either of the forms a, k. a, where a represents a units of length, a being either an integer or of the form m/n, where m, n are both integers, and k is an integer or of the form m/n (where both m, n are integers) but not square. In other words, p, σ may have either of the forms a or A, where A represents A units of area and A is integral or of the form m/n, where m, n are both integers. It has been the habit of writers to give a and Ja as the alternative forms of p, but I shall always use A for the second in order to keep the dimensions right, because it must be borne in mind throughout that p is an irrational straight line.

As Euclid extends the signification of rational (pnrós, literally expressible), so he limits the scope of the term aλoyos (literally having no ratio) as applied to straight lines. That this limitation was started by himself may perhaps be inferred from the form of words "let straight lines incommensurable with it be called irrational." Irrational straight lines then are with Euclid straight lines commensurable neither in length nor in square with the assumed rational straight line. k. a where k is not square is not irrational; k. a is irrational, and so (as we shall see later on) is (√k + √λ) a.

DEFINITION 4.

Καὶ τὸ μὲν ἀπὸ τῆς προτεθείσης εὐθείας τετράγωνον ῥητόν, καὶ τὰ τούτῳ σύμμετρα ῥητά, τὰ δὲ τούτῳ ἀσύμμετρα ἄλογα καλείσθω, καὶ αἱ δυνάμεναι αὐτὰ ἄλογοι, εἰ μὲν τετράγωνα εἴη, αὐταὶ αἱ πλευραί, εἰ δὲ ἕτερά τινα εὐθύγραμμα, αἱ ἴσα αὐτοῖς τετράγωνα ἀναγράφουσαι.

As applied to areas, the terms rational and irrational have, on the other hand, the same sense with Euclid as we should attach to them. According to Euclid, if p is a rational straight line in his sense, p2 is rational and any area commensurable with it, i.e. of the form kp2 (where k is an integer, or of the form m/n, where m, n are integers), is rational; but any area of the form k. p is irrational. Euclid's rational area thus contains A units of area, where A is an integer or of the form m/n, where m, n are integers; and his irrational area is of the form k. A. His irrational area is then connected with his irrational straight line by making the latter the square root of the