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Definitions

Propositions

Notes on Translations
 

Definitions

Brief comments

1 Similar rectilinear figures are those that have the angles equal, one by one, and the sides about the equal equals proportional.
2. and reciprocal figures are when in each of the figures leaders and followers are ratios. This definition is somewhat opaque. It is dubiously a part of the text. Nonetheless, it had meaning to whoever wrote it. Obviously, the intention must be that the corresponding parts of figures are in inverse ratio; that is, the leader in one ratio is the follower in the other. So if A corresponds in one figure to a in the other, and B to b, A : Bb : a, so that Ais the leader in the first ratio and a the follower in the other. We might call the first a leading ratio in respect of A, a and the second a following ratio in respect of A, a.
3. A straight line is said to be cut in extreme and mean ratio when it is as the whole to the larger section, so the larger to the smaller. The construction has already taken place at II 11, but will also be accomplished at VI 30.
4. An altitude of any figure is the perpendicular drawn from the vertex to the base.  
5. A ratio is said to be composed from ratios when the sizes of the ratios being multiplied times themselves make some ratio/size(?). This definition is unlikely to a part of the Elements. Additionally, it suggests a tradition that treats ratios as numbers, Heron, Diophantus, etc. What is significant is that it is a Greek definition that involves the notion of the size of the ratio, an important concept in Medieval and Renaissance reductions of ratio to number.

Propositions:

Prop. 1: Triangles and parallelograms which are under the same height are to one another as the bases.

Prop. 2: If some straight-line is drawn parallel to one of the sides of a triangle it will cut the sides of the triangle in-ratio; and if the sides of a triangle are cut in-ratio, the straight-line joining at the sections will be parallel to the remaining side of the triangle.

Prop. 3: If the angle of a triangle is bisected, but the straight-line cutting the angle also cuts the base, the segments of the base will have the same ratio as the remaining sides of the triangle; and if the segments of the base have the same ratio as the remaining sides, the straight-line joined from the vertex to the section will bisect the angle of the triangle.

Prop. 4: The sides of equiangular triangles are in-ratio, and those subtending the equal angles homologous.

Prop. 5: If two triangles have their sides in-ratio, the triangles will be equiangular and will have those angles equal which the homologous sides subtend.

Prop. 6: If two triangles have one angle equal to one angle and the sides about the equal angles in-ratio, the triangles will be equiangular and have those angles equal which the corresponding sides subtend.

Prop. 7: If two triangles have one angle equal to one angle, the sides about other angles in-ratio, and each of the remaining angles either both smaller than or not smaller than a right-angle, the triangles will be equiangular and will have their angles be equal about which the sides are in-ratio.

Prop. 8: If in a right-angle triangle a perpendicular is drawn from the right angle to the base, the triangles at the perpendicular will be similar to the whole and to each other.

Corollary: It is, in fact, obvious from this that if in a right-angled triangle a perpendicular is drawn from the right angle to the base, the drawn straight-line is a mean in-ratio of the segments of the base, , just what it was required to show. {And furthermore, the side at any one of the segments is a mean in-ratio of the base and the segment.}

Prop. 9: Fom the given straight-line to cut off the prescribed part.

Prop. 10: To cut the given uncut straight-line similarly to the given cut straight-line.

Prop. 11: Given two straight-lines, to find additionally a third in-ratio.

Prop. 12: Given three straight-lines, to find additionally a fourth in-ratio.

Prop. 13: Given straight-lines to find additionally a mean in-ratio.

Prop. 14: The sides about the same angles of equal and equiangular parallelograms reciprocate; and those equal-angled parallelograms whose sides about equal angles reciprocate are equal.

Prop. 15: The sides about the equal angles of equal triangles having one angle equal to one reciprocate, and for triangles having one angle equal to one whose sides about the equal angles reciprocate, those triangles are equal.

Prop. 16: If four straight-lines are in-ratio, the rectangle enclosed by the extremes is equal to the rectrangle enclosed by the middles; and if the rectangle enclosed by the extremes is equal to rectangle enclosed by the middles, the four straight-lines will be in-ratio.

Prop. 17: If three straight-lines are in-ratio, the rectangle enclosed by the extremes is equal to the square from the mean. And if the rectangle enclosed by the extremes is equal to the square from the mean, the three straight-lines will be in-ratio.

Prop. 18: From the given straight-line to the given rectilinear-figure to describe up a both similar and similarly positioned rectilinear-figure.

Prop. 19: Similar triangles are to one another in a ratio duplicate of the corresponding sides.

Corollary: It is, in fact, obvious from this, that if three straight-lines are proportional, it is as the first to the third, so the form from the first to the second that's similar and similarly described up,

Prop. 20: Similar polygons are divided into similar triangles both into ones equal in plêthos and corresponding in ratios to the wholes, and the polygon to the polygon has a duplicate ratio of the corresponding side to the corresponding side.

Corollary: But in the same way it will also be shown in the case of similar quadrilaterals that they are in duplicate ratio of the corresponding sides.

Prop. 21: Figures similar to the same rectilinear-figure are also similar to one another.

Prop. 22: If four straight-lines are in-ratio, the rectilinear-figures that are similar and similarly described up from them are in-ratio and the straight-lines themselves will be in-ratio.

Prop. 23: Equiangular parallelograms have a ratio to one another that's composed from the sides.

Prop. 24: The parallelograms about the diameter of every parallelogram are similar both to the whole and to each other.

Prop. 25: To construct a similar rectilinear-figure to the given rectilinear-figure and equal to another that’s given.

Prop. 26: If from a parallelogram a parallelogram similar to the whole and being positioned similarly, having a common angle with it, is taken out, it is about the same diameter as the whole.

Prop. 27: Of all the parallelograms applied along the same straight-line and deficient by parallelogram forms similar and similarly positioned to that described up from the half straight-line, the {parallelogram} applied from the half straight-line, being similar to the remainder, is the largest.

Prop. 28: Along the given straight-line to apply a parallelogram equal to the given rectilinear-figure similar to a parallelogram form that’s given, deficient by a similar parallelogram form. But it is required the given rectilinear-form {to which it is required to apply an equal} not be larger than that described up from the half straight-line, similar to the remainder {that both from the half and that similar to which it is required it to be deficient}

Prop. 29: Along the given straight line to apply a parallelogram equal to the given parallelogram exceeding by a similar parallelogram form that’s given.

Prorp. 30: To cut the given limited straight-line in extreme and mean ratio.

Prop. 31: In right-angled triangles, the species from the sides subtending the right-angle is equal to the similar species from the sides enclosing the right angle that are and similarly inscribed up.

Prop. 32: If two triangles are composed at one angle having sides in-ratio to two sides so that the corresponding sides of them are also parallel, the remaining sides of the triangles will be on a straight line.

Prop. 33: In equal circles the angles have same ratio as the circular-arcs on which they stand, whether they stand at the centers or at the circular-arcs.

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Notes on translations:

plêthos is not translated. In Euclid's Elements, it is any collection of countable things, as opposed to an arithmos, which is a plêthos of units. Intuitively, a, b, g, d, e is equal to A, B, G, D, E in plêthos if there is a one to one correspondance between them. They are not said to be equal in arithmos, because neither is an arithmos.

ὁμόλογος--corresponding: literally, “in agreement.” Heath rightly argues that the term comes from the sense of the verb, ”to be in agreement” snd that it should be translated as ”corresponding.” Even so, at vi 20, he finds himself compelled to render, ”homologous with the wholes,”as “in the same ratios as the wholes,” while a line later rendering, “homologous sides” as ”corresponding sides.” See v notes on definition 11. It seems clearer to stick with Heath's choice than to use the more literal but less familiar “homologous.”

εἶδος--form. It is difficult to know how to translate this term of art in 4th cent. Greek philosophy, whether or not it is transposed to mathematical usage from there. Heath (English), Vitrac (French), Peyrard (Latin and French), Heiberg (Latin), and Thaer (German) use “figure,” “figura,” or “Figur,”the same word they use for σχῆμα. Simson translates εἶδος as “figure” and σχῆμα here as “scheme,” but in book i as “figure.” Exceptions are Acerbi (Italian) and Taisbak (English in his translation of the Data although it is possible that he would translate the term differently in the Elements), who translate εἶδος as “forma,” “form” and σχῆμα as “figura,” “figure.” This is not a small piece of pedantry. In the Data, the form (εἶδος) is a single form or shape as such, which can occur in multiple individuals. However, the figure (σχῆμα) is an individual, not repeatable, if copyable. A translation of Data def. 3 would make no sense without marking this distinction (trans. Taisbak): “Rectilineal figures are said to be given in form if each angle is given and the ratios of the sides to one another are given.” We might put it, anachronistically, that the form constitutes a similarlty class, but there is no need to. The discussions of forms in 4th cent Greek philosophy provide enough alternatives to constitute a range of treatments of qualities applied to individuals. Without committing Euclid to any of these or to anything more than the general notion of a quality that is 'predicated’ of many things' or exemplified in many things, in books vi and x, where the book vi applications of areas get applied, we can always think of the form in this very way. It is then enough that we should be able to speak of a parallelogram as an individual mathematical figure and to speak of a parallelogram qua shaped or having shape. Again, one has but to look, for example, at the many interpretations of Aristotle, Categories 8 to get a sense of the range of possibilities. I should note that Mugler, Dictionnaire historique de la terminologie géométrique des grec, thinks that εἶδος has different meanings in the Elements and in the Data (so that Taisbak's choice would not illuminate the Elements). I think that a careful reading of the Elements makes this unnecessary and that it would separate the two treatises more than one would want.

ἀνάλογον--in-ratio. Surely the expression means, expanded, "at a ratio apiece,” on the analogy of ἀνά τὰ 5, meaning, “at a rate of 5 apiece.” The issue of how to translate the abstract noun, ἀναλογία, does not arise in book vi.

Translations of ratios explained: the most common way of expressing a ratio in the Elements is literally and ungrammatically: is so A to B thus so G to D (ἐστὶν ὡς τὸ Α πρὸς τὸ Β οὕτως τὸ Γ πρὸς τὸ Δ). It is natural to ask whether the copula “is” goes with either clause. The most natural would be to have it go with the second clause, as Euclid will sumetimes insert a different verb there, e.g. “was supposed,” so that we might translate, “as A to B so G is to D.” However, this loses something valuable in Euclid's phrasing, which isolates the elements of the proportion with 5 connecting words in order, the introductory copula, a marker the first ratio, one that marks the second, with each ratio the terms of the ratio separated by the two tokens of the preposition. This is as close as Euclid gets to a straight-forward mathematical notation, which the English reader might also appreciate. So I translate tersely, “It is: as A to B so G to D,” where the phrase, “It is,” marks the start of the proportion.

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