| and if | a > c, then d > f, |
| but if | a = c, then d = f, |
| and if | a < c, then d < f. |
By aid of these two propositions the following two are proved.
§ 55. Prop. 22. If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in order, the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last.
We may state it more generally, thus:
then not only have two consecutive, but any two magnitudes on the first side, the same ratio as the corresponding magnitudes on the other. For instance—
Prop. 23 we state only in symbols, viz.:—
| If | a : b : c : d : e : ... = 1/a′ : 1/b′ : 1/c′ : 1/d′ : 1/e′ . . . , | |
| then | a : c = c′ : a′, | |
| b : e = e′ : b′, | ||
and so on.
Prop. 24 comes to this: If a : b = c : d and e : b = f : d, then
Some of the proportions which are considered in the above propositions have special names. These we have omitted, as being of no use, since algebra has enabled us to bring the different operations contained in the propositions under a common point of view.
§ 56. The last proposition in the fifth book is of a different character.
Prop. 25. If four magnitudes of the same kind be proportional, the greatest and least of them together shall be greater than the other two together. In symbols—
If a, b, c, d be magnitudes of the same kind, and if a : b = c : d, and if a is the greatest, hence d the least, then a + d > b + c.
§ 57. We return once again to the question. What is a ratio? We have seen that we may treat ratios as magnitudes, and that all ratios are magnitudes of the same kind, for we may compare any two as to their magnitude. It will presently be shown that ratios of lines may be considered as quotients of lines, so that a ratio appears as answer to the question, How often is one line contained in another? But the answer to this question is given by a number, at least in some cases, and in all cases if we admit incommensurable numbers. Considered from this point of view, we may say the fifth book of the Elements shows that some of the simpler algebraical operations hold for incommensurable numbers. In the ordinary algebraical treatment of numbers this proof is altogether omitted, or given by a process of limits which does not seem to be natural to the subject.
Book VI.
§ 58. The sixth book contains the theory of similar figures. After a few definitions explaining terms, the first proposition gives the first application of the theory of proportion.
Prop. 1. Triangles and parallelograms of the same altitude are to one another as their bases.
The proof has already been considered in § 49.
From this follows easily the important theorem
Prop. 2. If a straight line be drawn parallel to one of the sides of a triangle it shall cut the other sides, or those sides produced, proportionally; and if the sides or the sides produced be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle.
§ 59. The next proposition, together with one added by Simson as Prop. A, may be expressed more conveniently if we introduce a modern phraseology, viz. if in a line AB we assume a point C between A and B, we shall say that C divides AB internally in the ratio AC : CB; but if C be taken in the line AB produced, we shall say that AB is divided externally in the ratio AC : CB.
The two propositions then come to this:
Prop. 3. The bisector of an angle in a triangle divides the opposite side internally in a ratio equal to the ratio of the two sides including that angle; and conversely, if a line through the vertex of a triangle divide the base internally in the ratio of the two other sides, then that line bisects the angle at the vertex.
Simson’s Prop. A. The line which bisects an exterior angle of a triangle divides the opposite side externally in the ratio of the other sides; and conversely, if a line through the vertex of a triangle divide the base externally in the ratio of the sides, then it bisects an exterior angle at the vertex of the triangle.
If we combine both we have—
The two lines which bisect the interior and exterior angles at one vertex of a triangle divide the opposite side internally and externally in the same ratio, viz. in the ratio of the other two sides.
§ 60. The next four propositions contain the theory of similar triangles, of which four cases are considered. They may be stated together.
Two triangles are similar,—
1. (Prop. 4). If the triangles are equiangular:
2. (Prop. 5). If the sides of the one are proportional to those of the other;
3. (Prop. 6). If two sides in one are proportional to two sides in the other, and if the angles contained by these sides are equal;
4. (Prop. 7). If two sides in one are proportional to two sides in the other, if the angles opposite homologous sides are equal, and if the angles opposite the other homologous sides are both acute, both right or both obtuse; homologous sides being in each case those which are opposite equal angles.
An important application of these theorems is at once made to a right-angled triangle, viz.:—
Prop. 8. In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.
Corollary.—From this it is manifest that the perpendicular drawn from the right angle of a right-angled triangle to the base is a mean proportional between the segments of the base, and also that each of the sides is a mean proportional between the base and the segment of the base adjacent to that side.
§ 61. There follow four propositions containing problems, in language slightly different from Euclid’s, viz.:—
Prop. 9. To divide a straight line into a given number of equal parts.
Prop. 10. To divide a straight line in a given ratio.
Prop. 11. To find a third proportional to two given straight lines.
Prop. 12. To find a fourth proportional to three given straight lines.
Prop. 13. To find a mean proportional between two given straight lines.
The last three may be written as equations with one unknown quantity—viz. if we call the given straight lines a, b, c, and the required line x, we have to find a line x so that
Prop. 11.
Prop. 12.
Prop. 13.
We shall see presently how these may be written without the signs of ratios.
§ 62. Euclid considers next proportions connected with parallelograms and triangles which are equal in area.
Prop. 14. Equal parallelograms which have one angle of the one equal to one angle of the other have their sides about the equal angles reciprocally proportional; and parallelograms which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.
Prop. 15. Equal triangles which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional; and triangles which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.
 |
The latter proposition is really the same as the former, for if, as in the accompanying diagram, in the figure belonging to the former the two equal parallelograms AB and BC be bisected by the lines DF and EG, and if EF be drawn, we get the figure belonging to the latter.
It is worth noticing that the lines FE and DG are parallel. We may state therefore the theorem—
If two triangles are equal in area, and have one angle in the one vertically opposite to one angle in the other, then the two straight lines which join the remaining two vertices of the one to those of the other triangle are parallel.
§ 63. A most important theorem is
Prop. 16. If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means; and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines are proportionals.
In symbols, if a, b, c, d are the four lines, and
| if | a : b = c : d, |
| then | ad = bc; |
| and conversely, if | ad = bc, |
| then | a : b = c : d, |
where ad and bc denote (as in § 20), the areas of the rectangles contained by a and d and by b and c respectively.
This allows us to transform every proportion between four lines into an equation between two products.
It shows further that the operation of forming a product of two lines, and the operation of forming their ratio are each the inverse of the other.
If we now define a quotient ab of two lines as the number which multiplied into b gives a, so that
