Commentary on Book VI Definitions¶
Back to Book VI Definitions.
DEFINITION 1.¶
Ὅμοια σχήματά ἐστιν εὐθύγραμμα, ὅσα τάς τε γωνίας ἴσας ἔχει κατὰ μίαν καὶ τὰς περὶ τὰς ἴσας γωνίας πλευρὰς ἀνάλογον.
This definition is quoted by Aristotle, Anal. post. II. 17, 99 a 13, where he says that similarity (τὸ ὅμοιον) in the case of figures “consists, let us say (ἴσως), in their having their sides proportional and their angles equal.” The use of the word ἴσως may suggest that, in Aristotle’s time, this definition had not quite established itself in the text-books (Heiberg, Mathematisches zu Aristoteles, p. 9).
It was pointed out in Van Swinden’s Elements of Geometry (Jacobi’s edition, 1834, pp. 114–5) that Euclid omits to state an essential part of the definition, namely that “the corresponding sides must be opposite to equal angles,” which is necessary in order that the corresponding sides may follow in the same order in both figures.
At the same time the definition states more than is absolutely necessary, for it is true to say that two polygons are similar when, if the sides and angles are taken in the same order, the angles are equal and the sides about the equal angles are proportional, omitting
(1) three consecutive angles,
or (2) two consecutive angles and the side common to them,
or (3) two consecutive sides and the angle included by them,
and making no assumption with regard to the omitted sides and angles.
Austin objected to this definition on the ground that it is not obvious that the properties (1) of having their angles respectively equal and (2) of having the sides about the equal angles proportional can co-exist in two figures; but, a definition not being concerned to prove the existence of the thing defined, the objection falls to the ground. We are properly left to satisfy ourselves as to the existence of similar figures in the course of the exposition in Book VI., where we learn how to construct on any given straight line a rectilineal figure similar to a given one (VI. 18).
DEFINITION 2.¶
The Greek text gives here a definition of reciprocally related figures (ἀντιπεπονθότα σχήματα). “[Two] figures are reciprocally related when there are in each of the two figures antecedent and consequent ratios” (Ἀντιπεπονθότα δὲ σχήματά ἐστιν, ὅταν ἐν ἑκατέρῳ τῶν σχημάτων ἡγούμενοί τε καὶ ἑπόμενοι λόγοι ὦσιν). No intelligible meaning can be attached to “antecedent and consequent ratios” here; the sense would require rather “an antecedent and a consequent of (two equal) ratios in each figure.” Hence Candalla and Peyrard read λόγων ὅροι (“terms of ratios”) instead of λόγοι. Camerer reads λόγων without ὅροι. But the objection to the definition lies deeper. It is never used; when we come, in VI. 14, 15, XI. 34 etc. to parallelograms, triangles etc. having the property indicated, they are not called “reciprocal” parallelograms etc., but parallelograms etc. “the sides of which are reciprocally proportional,” ὧν ἀντιπεπόνθασιν αἱ πλευραί. Hence Simson appears to be right in condemning the definition; it may have been interpolated from Heron, who has it.
Simson proposes in his note to substitute the following definition. “Two magnitudes are said to be reciprocally proportional to two others when one of the first is to one of the other magnitudes as the remaining one of the last two is to the remaining one of the first.” This definition requires that the magnitudes shall be all of the same kind.
DEFINITION 3.¶
DEFINITION 4.¶
Ὕψος ἐστὶ παντὸς σχήματος ἡ ἀπὸ τῆς κορυφῆς ἐπὶ τὴν βάσιν κάθετος ἀγομένη.
The definition of “height” is not found in Campanus and is perhaps rightly suspected, since it does not apply in terms to parallelograms, parallelepipeds, cylinders and prisms, though it is used in the Elements with reference to these latter figures. Aristotle does not appear to know altitude (ὕψος) in the mathematical sense; he uses κάθετος of triangles (Meteorologica III. 3, 373 a 11). The term is however readily understood, and scarcely requires definition.
[DEFINITION 5.¶
Λόγος ἐκ λόγων συγκεῖσθαι λέγεται, ὅταν αἱ τῶν λόγων πηλικότητες ἐφ’ ἑαυτὰς πολλαπλασιασθεῖσαι ποιῶσί τινα.
“A ratio is said to be compounded of ratios when the sizes (πηλικότητες) of the ratios multiplied together make some (? ratio, or size).”]
As already remarked (pp. 116, 132), it is beyond doubt that this definition of ratio is interpolated. It has little ms. authority. The best ms. (P) only has it in the margin; it is omitted altogether in Campanus’ translation from the Arabic; and the other mss. which contain it do not agree in the position which they give to it. There is no reference to the definition in the place where compound ratio is mentioned for the first time (VI. 23), nor anywhere else in Euclid; neither is it ever referred to by the other great geometers, Archimedes, Apollonius and the rest. It appears to be only twice mentioned at all, (1) in the passage of Eutocius referred to above (p. 116) and (2) by Theon in his commentary on Ptolemy’s σύνταξις. Moreover the content of the definition is in itself suspicious. It speaks of the “sizes of ratios being multiplied together (literally, into themselves),” an operation unknown to geometry. There is no wonder that Eutocius, and apparently Theon also, in their efforts to explain it, had to give the word πηλικότης a meaning which has no application except in the case of such ratios as can be expressed by numbers (Eutocius e.g. making it the “number by which the ratio is called”). Nor is it surprising that Wallis should have found it necessary to substitute for the “quantitas” of Commandinus a different translation, “quantuplicity,” which he said was represented by the “exponent of the ratio” (rationis exponens), what Peletarius had described as “denominatio ipsae proportionis” and Clavius as “denominator.” The fact is that the definition is ungeometrical and useless, as was already seen by Savile, in whose view it was one of the two blemishes in the body of geometry (the other being of course Postulate 5).
It is right to add that Hultsch (art. “Eukleides” in Pauly-Wissowa’s Real-Encyclopädie der classischen Altertumswissenschaft) thought the definition genuine. His grounds are (1) that it stood in the παλαιὰ ἔκδοσις represented by P (though P has it in the margin only) and (2) that some explanation on the subject must have been given by way of preparation for VI. 23, while there is nothing in the definition which is inconsistent with the mode of statement of VI. 23. If the definition is after all genuine, I should be inclined to regard it as a mere survival from earlier textbooks, like the first of the two alternative definitions of a solid angle (XI. Def. 11); for its form seems to suit the old theory of proportion, applicable to commensurable magnitudes only, better than the generalised theory of Eudoxus.