tions.

(a) If A: C:: A': C', and B: C:: B': C', then At B:C:: A'+ B' : C'; and so, if any number of proportions have the same consequents 54 (b) If A: B: A': B', and B: C:: B': C', then, ex æquali, A: C:: A': C'; and so, if there are any number of magnitudes A, B, C, D, and A', B', C', D', the ratios of which are the same respectively in the same order. 54 (c) If A: B:: B': C', and B: C:: A B', then ex æquo perturbato, AC: A': C'; and so, if there are any number of magnitudes A, B, C, D, and A', B', C', D', the ratios of which are the same respectively in

a cross order.

55

(D) Rules of Arithmetical Proportion. (a) If four magnitudes are proportionals, and if A, B, C, D represent those magnitudes numerically, then

(d) The same being supposed A× M:
B::CxM: D, and conversely
sch. 47
(e) If four magnitudes, numerically re-
presented by A, B, C, D, are propor-
tionals;
and four others, represented

by A, B, C, D', are likewise proportionals; then Ax A': B x B':: CX CD x D'; a rule which is called "compounding the proportions," and is applicable to any number of proportions sch. 48 Proportional. See "Third Proportional," "Fourth Proportional," "Mean Proportional."

Proportional Straight Lines. See "Straight

Line."

Pyramid, (also its vertex, base, sides, principal edges, lateral or convex surface, frustum, or truncated pyramid) def. 127 When said to be regular (its axis) When two pyramids are said to be similar

When symmetrical. See "Polyhedron."

127

126

(a) Triangular pyramids which have equal bases and altitudes are equal to 145 one another.

(6) Every pyramid (triangular or other

and this will be the case, how great soever the

md numbers m and n may be taken; that is, how great soever the numbers nb and m d may be. But beq md

A

=

B

cause

P differs from n b

na nb 1 md

1 and by less than nb

by increasing m and n,

four are proportionals*

This is true, not only when the magnitudes are commensurable proportionals (as is demonstrated in p. 47), but also when four magnitudes are proportionals according to def. 7, and as such, is meant to be asserted by placing it among the properties of proportionals. It is necessary, therefore, to add the following demonstration of the general case:

Let A, B, C, D, be proportionals according to def. 7, p. 49; let B be divided into any number, b, of equal parts, and let A contain a of those parts; that is, a certain number exactly together with a fraction, which number and fraction are, together, represented by a; also let D be divided into any number, d, of equal parts, and let C contain c of those parts, c being a whole number and fraction as before;

by less than any the same given difference.

Р Therefore, because is always equal to nb

१ md

na nb

Also, conversely, if

=

b

, A, B, C, D shall be

proportionals according to def. 7. For let B and D of equal parts; be divided, each, into any number, n,

a

1

th

then, because A contains ths of B, it contains n

of B as often as the number n is contained in 1

a

and in like manner C contains

th of D as often as

na nb

-ths of B, it contains ths of B, and in like

n is contained in

; but, because

=

en is con

wise) is equal to the third part of a prism which has the same base and altitude. 146, 147 (c) The solid content of a pyramid is equal to one-third of the product of its base and altitude cor. 147 (d) The frustum of a pyramid is equal to the sum of three pyramids which have the same altitude with the frustum, and for their bases the sum of its two bases and a mean proportional between them 147 (e) Pyramids which have equal altitudes are to one another as their bases, and pyramids which have equal bases, as their altitudes; also, any two pyramids are to one another in the ratio which is compounded of the ratios of their bases and altitudes cor. 147 [() Pyramids which have for their bases similar polygons, and the principal edges drawn from two corresponding angles of those polygons making equal angles with each of the homologous adjoining sides, and in the same ratio as those sides, are similar.] (g) Similar pyramids are to one another in the triplicate ratio (or as the cubes) of their homologous edges [(h) Symmetrical pyramids are equal to one another; for they have equal bases and altitudes.]

149

If the convex surface of a pyramid be produced to any extent, the sections of it by parallel planes will be similar polygons

sch. 145

Quadrature of the circle; an exact quadrature is impracticable in the way of calculation, sch. 99. See "Circle" and "Lunes." Quadrilateral plane figure

def. 2

cor. 15

(a) Its angles are together equal to four right angles. (b) If any other than a parallelogram, the squares of its diagonals are together greater the squares of the four sides by four times the straight line which joins the middle point of the diagonals.

23

(c) If two quadrilaterals have three sides of the one equal to three sides of the other, each to each, and the angles of the one lying in the circumference of a circle of which the fourth side is diameter, but the angles of the other not so lying, the first quadrilateral shall be greater than the other 103 (d) If two quadrilaterals have the sides of the one equal to the sides of the other each to each, and the angles of the one lying in the circumference of a circle, but the angles of the other not so lying, the first quadrilateral shall be greater than the other 104 (B) (e) To construct a quadrilateral which shall have its sides equal to four given

These propositions (c) and (d) are true, whether the equal sides lie in the same, or in different orders.

def. 3, 127

31

Is independent of the kind of magnitudes compared, but requires that the two shall be of the same kind Of commensurable magnitudes, how expressed in numbers 32

31

Of incommensurable magnitudes. 48 Ratio of equality, inverse or reciprocal

ratio

def. 32 Equal ratios, greater ratio, less ratio def. 32, 49 Duplicate ratio, triplicate ratio, &c. (Also Sub-duplicate, Sub-triplicate, &c.) def. 34 Compound ratio; i. e., a ratio which is compounded of any number of ratios def. 34 When one ratio is said to be taken away from another

Of equal Ratios*.

45

(a) If A = B, then A: C = B: C; and conversely

51

51

(b) If A B C D and C:D= E F, then A: B E: F

(c) If A: BA': B', then A B AA: B±B'

53, 54 (d) If A: BA': B'A′′ : B′′, then A B A+ A+ A B+ B'B'; and so of any number of equal ratios 54

(e) Magnitudes are to one another as their equimultiples

52

cor. 52

(ƒ) If A ; B = C: D, mA; mB = nCnD; and conversely (g) If A: B = C: D, mA:n C = mCnD; and conversely 52 (h) If there are any number of magnitudes of the same kind A, B, C, D, and as many others A', B', C', D', and if the ratios of the first to the second, of the second to the third, of the third to the fourth, and so on, are the same respectively in the two series, any two combinations by sum and difference of the magnitudes in the first series shall be to one another as two similar combinations of the magnitudes in the second series cor. 55 Of unequal Ratios.

(a) Ratios which are compounded of the
same ratios, in whatsoever orders, are
equal to one another
55
(b) Ratios which are the duplicates, tri-
plicates, &c. of equal ratios, are equal
to one another
cor. 56

(c) If equal ratios are taken from equal
ratios, the remaining ratios are equal
cor. 56

(D) Of reciprocal Ratios.

40

(a) If a ratio, which is compounded of two ratios be a ratio of equality, one of these is the reciprocal of the other; and conversely cor. 51 (b) The reciprocals of equal ratios are equal (c) In the composition (i. e., compounding) of ratios, any two which are reciprocals may be neglected without affecting the resulting compound ratio cor. 56 See also"Numerical Ratio" and "Proportion." Reciprocal, one ratio said to be of another 32 Reciprocally proportional; two magnitudes

and other two are said to be reciprocally proportional, when the four constitute a proportion, in which one pair are extremes and the other means. Rectangle def. 2 [(a) Its diagonals are equal to one another, and all its angles are right angles (I. 4.).]

(b) If the adjoining sides of a rectangle contain any the same straight line M, the one a times and the other b times, the rectangle shall contain the square of M, a xb times

18

(c) Hence, a rectangle is measured by,

This (although only demonstrated in p. 18, in the case of whole numbers) is true generally, i. e. when a and b are any numbers whatever.

Of the general case we add the following demonstration-Let CA, C B be the sides of the rectangle, containing Ma and b times respectively; and, first, let a be supposed a whole number. Let the square of M be contained in the rectangle a x d times; and, first, let d be supposed greater than b, so that if C D be taken equal to d times M, it will be greater than C B, which is only b times M. Let any part of M, which is less than BD, say the half, be taken, and

(f) Rectangles, which have their sides reciprocally proportional, are equal to one another; and, conversely, rectan gles, which are equal to one another, have their sides reciprocally propor tional 63 (9) The equality of two rectangles is convertible into a proportion of four straight lines; and vice versa sch.64 See" Straight Line" and "Parallelogram." Rectangular Parallelopiped def. 126 [(a) Its diagonals are equal to one another, and every face is at right angles to the two adjoining faces.] (b) If the conterminous edges of a rectangular parallelopiped contain any the same straight line M, a, b, and e times respectively, the rectangular parallelopiped shall contain the cube of M, a xbx c times +

142

Mis contained in C A 2a times, and in CD 24 times, and in CE an exact number of times (say p) less than 2d, its square is contained in the rectangle ACEF 2axp times, that is, less than 2ax 2d times; but its square is contained in the rectangle A B 2ax 2d times, for the square of M is supposed to be contained in A Baxdtimes; therefore the rectangle AB is greater than A ECF, which is impossibleconsequently d cannot be greater than b. And in the same manner it may be shown that it cannot be less than b, that is, it is equal to it. And this being shown in the case in which one of the numbers, as a, is whole, the like demonstration, resting upon this, may be applied to the case in which neither of them is a whole number.

The case of the rectangular parallelopiped may be demonstrated in the same manner.

This is true, whatever the numbers a, b, and e are-see the last note.,

cor. 8

(A) (a) Any one of the sides is less than the sum of all the others (b) The exterior angles of a rectilineal figure (formed by producing its sides) are together equal to four right angles; and the sum of its interior angles, together with four right angles, is equal to twice as many right angles as the figure has sides (c) The area of a rectilineal figure may be obtained by dividing it into triangles, having for their common vertex one of the angular points of the figure, or some point within it sch. 18

(B) Of similar rectilineal figures.

15

(a) Rectilineal figures of more than three sides which have their several angles equal in order, each to each, are not for that reason necessarily similar sch, 59 (b) Rectilineal figures which have all their angles but two, equal each to each in order, and the sides about the equal angles proportionals, are similar cor. 60

(c) Any two which are similar to the same rectilineal figure are similar to one another def. 57 (d) Similar rectilineal figures are divided into the same number of similar triangles by the diagonals which are drawn from any two corresponding angles cor. 60 (e) Lines, which are similarly placed in similar figures, cut the homologous sides at equal angles, and are to one another as those sides cor. 60

(f) Similar rectilineal figures are to one another in the duplicate ratio (or as the squares) of their homologous sides; and their perimeters are as those sides] 67

(g) If four straight lines are proportionals, any two similar rectilineal figures which have the first and second for homologous sides, and any two which have the third and fourth, are proportionals cor. 67 (h) If similar rectilineal figures are similarly described upon the hypotenuse and sides of a right angled triangle, the figure upon the hypote nuse is equal to the sum of the figures upon the two sides

67

(C) Of rectilineal figures which are not

similar.

(a) If a rectilineal figure have not all

(D)

(c) A regular polygon is greater than any other rectilineal figure which has the same (or a less) number of sides, and the same perimeter cor. 101, 102 (d) A regular polygon has a less perimeter than any rectilineal figure which has the same (or a less) number of sides and the same area cor. 101 (e) If a rectilineal figure have not all its angles in the circumference of a circle, a greater may be found having the same sides • cor. 105 (f) A rectilineal figure, which has all its angles lying in the circumference of a circle, is greater than any other having the same sides, whether in the same or in different orders cor. 105 (g) If two rectilineal figures have all their sides but one equal each to each (in the same, or in different orders), and the angles of the one lying in the circumference of a circle of which the excepted side is the diameter; but the angles of the other not so lying, the first figure shall be greater than the other note 103

Regular Polygon, def. (also its apothem and centre)

91