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THEOREM LXVI,

Equimultiples of any two Quantities have the same Ratio as

the Quantities themselves. LET A and B be any'two quantities, and mA, MB, any equimultiples of them; m being any nunber whatever : then will mA and mB have the same ratio as A and B, or

A: B :: MA : mB.

A

MB

B For

the same ratio. MA Corol. Hence, like parts of quantities have the same ratio as the wholes; because the wholes are equimultiples of the like parts, or A and B are like parts of mA and mB.

THEOREM LXVII.

IF Four Quantities, of the Same Kind, be Proportionals s

they will be in Proportion by Alternation or Permutation, or the Antecedents will have the Same Ratio as the Consequents. LET A : B :: ma : MB; then will A : MA :: B : MB,

MA For

m,

both the same ratio,

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m, and

B

THEOREM LXVIII.

if Four Quantities be Proportional ; they will be in Pro

portion by Inversion, or Inversely, LET A:B :: MA: MB; then will B A :: mB: MA,

ma For

both the same ratio.

A

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B

THEOREM LXIX.

If Four Quantities be Proportional; they will be in Pro

portion by Composition and Division.
LET A:B :: MA : MB;
Then will B + A: A :: MB + mA : MA,

and B + A:B :: MB £ inA : inB.
ТА
А

B
For,
mB Ima B IA mB MA

Corol.

INB

“; and

Cord. It appears from hence, that the Sum of the Greatest and Least of four proportional quantities, of the same kind, exceeds the Sum of the Two Means. For, since A: A + B :: MA : MA + MB, where A is the least, and mA + mB the greatest; then n + 1.A + mB, the sum of the greatest and least, cxceeds ni +1.A + B the sum of the two means.

THEOREM LXX.

If, of Four Proportional Quantities, there be taken any

Equimultiples whatever of the two Antecedents, and any Equimultiples, whatever of the two Consequents ; the quantities resulting will still be proportional.

LET A : B': : 01A : B; also, let pa and pma be any equimultiples of the two antecedents, and qB and qmB any equimultiples of the two consequents; then will pa :qB :: pin A : qmB. For G17B qB

both the same ratio. pma

PA '

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THEOREM LXXI.

If there be Four ?'roportional Quantities, and the two

Consequents be either Augmented or Diminished by Quantities that have the Same Ratio as the respective Antecedents; the Results and the Antecedents will still be Proportionals.

LET A:B ::A : mB, and nA and nma any two quantities having the sai; e ratio as the two antecedents; then will A:B INA ::mi : IB + nmA.

mB # nma BIA For

both the same ratio.

MA

A

THEOREM LXXII.

If any Number of Quantities be Proportional, then any

one of the Antecedents will be to its Consequent, as the Sum of all the Antecedents is to the Sum of all the Consequents.

LET A:B :: ma : MB ::na : nB, &c; then will A :B :: A + ma + na ::8 + mB + nB, &c.

B + mB + nB For

the same ratio. Atma +

THEOREM

B

THEOREM LXXIII.

If a Whole Magnitude be to a Whole, as a Part taken from

the first, is to a Part taken from the other, then the Remainder will be to the Remainder, as the whole to the whole.

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If any Quantities be Proportional; their Squares, or Cubes,

or any Like Powers, or Roots, of them, will also be Pro

portional LET A :B::ma : MB; then will a":B" :: m'a": m"B". m"B"

B For

both the same ratio. in" A"

THEOREM LXXV.

If there be two Sets of Proportionals; then the Products or

Rectangles of the Corresponding Terms will also be Proportional. LET A:B :: MA': MB; and c:D::nc : ND ; then will AC: BD :: MNAC : MNBD.

пnBD For

both the sane ratio. mnAC

BD

AC

THEOREM LXXVI.

If Four Quantities be Proportional; the Rectangle or Product

of the two Extremes, will be Equal to the Rectangle or Product of the two Means. And the converse. LET A:B :: MA : MB; then is A X MB =BX mA = MAB, as is evident.

THEOREM

THEOREM LXXVIT.

If Three Quantities be Continued Proportionals; the Rect

angle or Product of the two Extremes, will be Equal to the Square of the Mean. And the converse. LET A, ma, m'a be three proportionals, or A:MA :: MA : m’A; then is a x m'a = m’A, as is evident.

THROREM LXXVIII.

If any Number of Quantities be Continued Proportionals;

the Ratio of the First to the Third, will be duplicate or the Square of the Ratio of the First and Second; and the Ratio of the First and Fourth will be triplicate or the cube of that of the First and Second ; and so on. LET A, ma, m’A, mA, &c, be proportionals;

MA then is

but
m2; and

= m;

&c. A

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mA

= m;

A

THEOREM LXXIX.

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Triangles, and also Parallelograms, having equal Altitudes,

are to each other as their Bases. LET the two triangles ADC, DEF, have CR the same altitude, or be between the same parallels AE, CF; then is the surface of the triangle Adc, to the surface of the triangle DEF, as the base ad is to the DGHI base DE.

Or, AD : DE :: the triangle ADC : the triangle DEF.

For, let the base Ad be to the base de, as any one num. ber m (2), to any other number n (3); and divide the respective bases into those parts, AB, BD, DG, GH, HE, all equal to one another; and from the points of division draw the lines BC, FG, FH, to the vertices c and F. Then will these lines divide the triangles ADC, DEF, into the: same number of parts as their bases, each equal to the triangle ABC, because those triangular parts have equal bases and altitude (corol. 2, th. 25); namely, the triangle ABC, equal to each of the triangles BDC, DFG, GFH, HFE.

So that the triangle ADC, is to the triangle Die, as the number of

parts

parts m (2) of the former, to the number n (3) of the latter, that is, as the base ap to the base de (def. 79).

In like manner, the parallelogram Adki is to the parallelogram ĐEFK, as the base Ad is to the base de; each of these having the same ratio as the number of their parts, un to n. Q. E. D.

THEOREMI LXXX,

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Triangles, and also Parallelograms, having Equal Bases, are

to each other as their Altitudes, LET ABC, BEF, be two triangles having the equal bases AB, BE, and

K к whose altitudes are the perpendiculars CG, FH; then will the triangle ABC: the triangle BEF :: CG : FH. For, let BK be perpendicular to AB,

В н and equal to cG; in which let there be taken BL = FH; drawing AK and AL.

Then, triangles of equal bases and heights being equal (corol. 2, th. 25), the triangle ABK is = ABC, and the triangle ABL = BEF. But, considering now ABK, ABL, as two triangles on the bases BK, BL, and having the same altitude AB, these will be as their bases (th. 79), namely, the triangle ABK: the triangle ABL :: BK : BL. But the triangle ABK = ABC, and the triangle ABL = BEF,

also BK = cG, and BL = FH. Theref, the triangle ABC : triangle BEF :: CG : FH.

And since parallelograms are the doubles of these triangles, having the same bases and altitudes, they will likewise have to each other the same ratio as their altitudes. Q. E. D.

Corol. Since, by this theorem, triangles and parallelograms, when their bases are equal, are to each other as their altitudes; and by the foregoing one, when their altitudes are equal, they are to each other as their bases; therefore universally, when neither are equal, they are to each other in the compound ratio, or as the rectangle or product of their bases and altitudes.

THEOREM

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