PROP. H. THEOREM.

If a ratio which is compounded of several ratios be equal to a ratio which is compounded of several other ratios; and if one of the first ratios, or the ratio which is compounded of several of them, be equal to one of the last ratios, or to the ratio which is compounded of several of them; the remaining ratio of the first, or, if there be more than one, the ratio compounded of the remaining ratios, is equal to the remaining ratio of the last, or, if there be more than one, to the ratio compounded of these remaining ratios.

Let the first ratios be those of A to B, B to C, C to D, D to E, and E to F; and let the other ratios be those of G to H, H to K, K to L, and L to M. Also let the ratio of A to F, compounded of the first ratios, be equal to the ratio of G to M, compounded of the other ratios. Besides, let the ratio of A to D, compounded of the ratios of A to B, B to C, and C to D, be equal to the ratio of G to K, compounded of the ratios of G to H, and H to K.

The ratio compounded of the remaining first ratios, viz., of the ratios of D to E, and E to F, which compounded ratio is the ratio of D to F, is equal to the ratio of K to M, which is com

pounded of the remaining ratios of K to L, and L

to M of the other ratios.

Because (Hyp.) A is to D, as G is to K.

There

fore, by inversion, D is to A, as K to G (V. B.).

A.B.C.D.E.F

G.H.K.L.M

Because A is to F, as G is to M (Hyp.). Therefore, ex æquali, D is to F, as K is to M (V. 22). If, therefore, a ratio which is, &c. Q. E. D.

If there be any number of ratios, and any number of other ratios such, that the ratio compounded of ratios equal to the first ratios, each to each, is equal to the ratio compounded of ratios which are equal, each to each, to the last ratios; and if one of the first ratios, or the ratio which is compounded of ratios equal to several of the first ratios, each to each, be equal to one of the last ratios, or to the ratio compounded of ratios which are equal, each to each, to several of the last ratios; the remaining ratio of the first, or, if there be more than one, the ratio compounded of ratios equal, each to each, to the remaining ratios of the first, is equal to the remaining ratio of the last, or, if there be more than one, to the ratio compounded of ratios which are equal, each to each, to these remaining ratios.

Let the ratios of A to B, C to D, and E to F, be the first ratios: and the ratios of G to H, K to L, M to N, O to P, Q to R, be the other ratios: and let A be to B, as S to T; and C to D, as T to V; and E to F, as V to X: and therefore (Def. A), the ratio of S to X is compounded of the ratios of S to T, T to V, and V to X, which are equal to the ratios of A to B, C to D, and E to F: each to each.

Also, let G be to H, as Y to Z: and K to L as Z to a; M to N as a to b; 0 to P, asb to c; and Q to R, as c to d: and therefore (Def. A), the ratio of Y to d is compounded of the ratios of Y to Z, Z to a, a to b, b to c, and c to d, which are equal, each to each, to the ratios of G to H, K to L, M to N, Ó to P, and Q to R: wherefore (Hyp.), S is to X, as Y to d.

с

Also, let the ratio of A to B, that is, the ratio of S to T, which is one of the first ratios, be equal to the ratio of e to g, which is compounded of the ratios of e to f, and f to g, which (Hyp.) are equal to the ratios of G to H, and K to L, two of the other ratios; and let the ratio of h to be that which is compounded of the ratios of h to k, and k to l, which are equal to the remaining first ratios, viz., of C to D, and E to F.

Also, let the ratio of m top, be that which is compounded of the ratios of m to n, n to o, and o to p, which are equal, each to each, to the remaining other ratios, viz., of M to N, O to P, and Q to R.

The ratio of h to l is equal to the ratio of m to p; that is, h is to l, as m is to p.

h, k, l.

A. B; G, H; K, L; e, f, g.

C, D; E, F,
M, N;. O, P, Q, R.
m, n, o, p.

S, T, V, X.

Y, Z; a, b, c, d.

Because e is to f, as G to H, that is, as Y to Z; and ƒ is to g, as K to L, that is, as Z to a. Therefore, ex æquali, e is to g, as Y to a (V. 22). But A is to B, that is, S is to T, as e is to g (Hyp.). Therefore S is to T, as Y is to a (V. 11). Wherefore, by inversion, T is to S, as a is to Y (V. B). But S is to X, as Y is to D (Hyp.). Therefore, ex equali, T is to X as a is to d. Because h is to k, as C is to D, that is, as Tis to V (Hyp.); and k is to las E is to F, that is, as V is to X. Therefore, ex æquali, h is to l, as T is to X. In like manner, it may be shown that m is to p, as a is to d. But it has been shown that T is to X, as a is to d. Therefore h is to l, as m is to p (V. 11). Wherefore, if there be any number of ratios, &c. Q. E. D.

"The propositions G and K are usually, for the sake of brevity, expressed in the same terms with propositions F and H: and therefore it was proper to show the true meaning of them when they are so expressed; especially since they are very frequently made use of by geometers."

After having laboured to correct this book of Euclid, Dr. Simson says, "I most readily agree with what the learned Dr. Barrow says, that there is nothing in the whole body of the Elements of a more subtile invention, nothing more solidly established, and more accurately handled, than the doctrine of proportionals.' And there is some ground to hope, that geometers will think that this could not have been said with as good reason, since Theon's time (A.D. 380), till the present." The modesty of this remark is only surpassed by its truth. Most editors, since Dr. Simson's time, have only rendered the doctrine of proportion more obscure; as a remarkable example, see Professor De Morgan's "Connection of Number and Magnitude."

BOOK VI.

DEFINITIONS.

I.

SIMILAR rectilineal figures are those which have their several angle equal, each to each, and the sides about the equal angles proportionals.

In the case of triangles, this definition is redundant. For it is proved in Prop. IV. of this Book, that the sides about the equal angles of equiangular triangles are proportionals. In the case of quadrilaterals, or polygons, however, the definition is necessary. According to this definition, all equilateral triangles, squares, and regular polygons are similar rectilineal figures.

II.

Triangles and parallelograms are said to have their sides reciprocally proportional, when the sides about two of their angles are proportionals in such a manner, that a side of the first figure is to a side of the second, as the remaining side of the second is to the remaining side of the first.

Two magnitudes of any kind may be said to be reciprocally proportional to other two of the same kind, when one of the first pair is to one of the second pair, as the remaining one of the second pair is to the remaining one of the first.

III.

A straight line is said to be cut in extreme and mean ratio, when the whole is to the greater segment, as the greater segment is to the less.

A straight line is said to be cut in harmonical ratio, when the whole is to one of the extreme segments as the other extreme segment is to the middle segment. For brevity's sake, this mode of dividing a straight line is called harmonical section; and the mode of dividing a straight line explained in Euclid's definition is called medial section.

IV..

The altitude of any figure is the straight line drawn from its vertex perpendicular to the base.

Altitude is a term synonymous with perpendicular. By the vertex of a figure here, is meant that angular point of the figure which is most remote from any side of the figure assumed at the base, the degree of remoteness being measured by a perpendicular drawn to that side or that side produced, from the said vertex; in other words, the longest perpendicular drawn from any angular point to the base, or base produced, is the altitude.

PROP. I. THEOREM.

Triangles and parallelograms of the same altitude are to one another as their bases.

Let the triangles ABC and A CD, and the parallelograms E C and CF, have the same altitude, viz., the perpendicular drawn from the

point A to BD or BD produced. As the base B C is to the base CD, so is the triangle ABC to the triangle A CD; and as the base BC is to the base CD, so is the parallelogram EC to

the parallelogram CF.

Produce BD both ways to the points H and L, and take any number of straight lines BG and GH, each equal to the base BC (I. 3); and DK and K L, any number of straight lines each equal to the base CD. Join AĞ, AH, AK, and AL.

E A F

HG BC D K

Because CB, B G and GH, are all equal, the triangles AHG, AGB and AB C, are all equal (I. 38). Therefore, whatever multiple the base HC is of the base B C, the same multiple is the triangle AHC of the triangle A B C. For the same reason, whatever multiple the base LC is of the base CD, the same multiple is the triangle ALC of the triangle ADC. But if the base HC be equal to the base CL, the triangle AHC is also equal to the triangle ALC (I. 38); and if the base HC be greater than the base CL, the triangle AHC is likewise greater than the triangle ALC; and if less, less. Because there are four magnitudes, viz., the two bases BC and C D, and the two triangles A B C and ACD; and of the base B C, and the triangle A B C, the first and the third, any equimultiples whatever have been taken, viz., the base HC and the triangle AHC; and of the base CD, and the triangle ACD, the second and the fourth, any equimultiples whatever have been taken, viz., the base CL and the triangle AL C. And it has been shown, that, if the base HC be greater than the base CL, the triangle AHC is greater than the triangle ALC; if equal, equal; and if less, less. Therefore, as the base B C is to the base CD, so is the triangle ABC to the triangle ACD(V.Def.5). Because the parallelogram CE is double of the triangle A B C (Ì. 41), and the parallelogram CF double of the triangle A CD, and magnitudes have the same ratio which their equimultiples have (V. 15). Therefore, as the triangle A B C is to the triangle ACD, so is the parallelogram EC to the parallelogram CF. But it has been shown, that, as the base B C is to the base CD, so is the triangle A B C to the triangle ACD. And as the triangle ABC is to the triangle ACD, so is the parallelogram E C to the parallelogram CF. Therefore, as the base B C is to the base CD, so is the parallelogram EC to the parallelogram CF (V. 11). Wherefore, triangles, &c. Q. E. D.

COROLLARY.-From this it is plain, that triangles and parallelograms that have equal altitudes, are to one another as their bases.

Let the figures be placed so as to have their bases in the same straight line; and having drawn perpendiculars from the vertices of the triangles to the bases, the straight line which joins the vertices is parallel to that in which their bases are (I. 33), because the perpendiculars are both equal and parallel to one another (I. 28). Then, if the same construction be made as in the proposition, the demonstration will be the same.

Exercise.-Triangles and parallelograms upon equal bases are to one another as their altitudes.

PROP. II. THEOREM.

If a straight line be drawn parallel to one of the sides of a triangle, it cuts the other two sides, or these sides produced, proportionally, so that the segments between the base and the parallel, are homologous; and conversely, if the two sides, or these sides produced, be cut proportionally, so that the segments between the base and the parallel are homologous, the straight line which joins the points of section is parallel to the base of the triangle.

Let DE be drawn parallel to B C, one of the sides of the triangle ABC. The sides A B and A C, or A B and AC produced, are cut proportionally; that is, BD is to DA, as CE is to E A.

The tri

A

D

B

A

E B

CD

E

D

E B

Join BE and CD. angle BDE is equal to the triangle CDE (I. 37), because they are on the same base DE, and between the same parallels DE, and B C. But ADE is another triangle; and equal magnitudes have the same ratio to the same magnitude (V. 7). Therefore, as the triangle BDE is to the triangle ADE, so is the triangle CDE to the triangle ADE. But the triangle BDE is to the triangle ADE, as BD is to DA (VI. 1), because their altitude is the perpendicular drawn from the point E to A B, and they are to one another as their bases. For the same reason, the triangle CDE is to the triangle AD E, as CE to E A. Therefore, as BD is to D A, so is CE to EA (V. 11).

Next, let the sides A B and AC of the triangle A B C, or A B and A C produced, be cut proportionally in the points D and E, that is, so that BD is to DA as CE to E A. Join DE, and it is parallel to B C.

The same construction being made, because BD is to DA as CE is to EA. But BD is to DA, as the triangle BDE is to the triangle ADE (VI. 1); and CE is to EA, as the triangle CDE is to the triangle ADE. Therefore the triangle BDE is to the triangle ADE, as the triangle CDE is to the triangle ADE (V. 11). Wherefore the triangles BDE and CDE have the same ratio to the triangle ADE. Therefore the triangle BDE is equal to the triangle CDE (V. 9); and they are on the same base D E. But equal triangles on the same base and on the same side of it, are between the same parallels (I. 39). Therefore DE is parallel to B C. Wherefore, if a straight line, &c. Q. E. D.

The necessity for three diagrams in this proposition arises from the variety of position which may be given to the straight line drawn parallel to the base. This parallel may be drawn between the vertex of the triangle and the base, as in the first figure; beyond the base, as in the second figure, when the sides must be produced through the extremities of the base to meet it; or, beyond the vertex, as in the third figure, when the sides must be produced through the vertex, to meet it. In all these cases, the demonstration holds equally good, and it should be read with each figure separately, in order to render the argument clear to the mind. A more general method of enunciating this proposition is contained in the corollary.

Corollary.-The triangles which two intersecting straight lines form with two parallel straight lines, have their sides on the intersecting lines proportionals, and those in the same straight line are homologous; and conversely, the two straight lines, which, with two intersecting straight lines, form triangles, having