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But the rectangle under the sum and difference of D E and E G, is equal to the rectangle under the sum and difference of D F and FG; (Theo. 48. Cor.) whence the rectangle of G B and G A is equal to the rectangle of G D and GC. Q. E. D.
Cor. 1. As the rectangle of AG and GB, the segments of the diameter is equal to the rectangle of GC and GD, the segments of any other line made by its intersection with the circle; it is evident that if any two lines drawn through the same point cut the circumference of a circle, the rectangle of the segments of the one will be equal to the rectangle of the segments of the other.
Cor. 2. If, when the point of meeting is without the circle, the line GCD revolve about the point G, till the points C and D coincide, as in H, GH will be a tangent to the circle ; and the rectangle of GC and G D will be equal to the square of GH; and hence also a tangent to the circle, drawn from G, on the other side of the diameter will be equal to the tangent G H.
THEOREM LXVIII. If B E F D be a circle, and from any point A in the diameter B D, or the diameter produced, the lines A E, A F, be drawn to any chord E F, which is parallel to B D, the squares of A E and AF together, will be equal to the squares of B A and AD, the segments of the diameter. From the centre C draw CG
E. G perpendicular to E F, and join
F AG and CE. Then E F is bisected in G (Theo. 60.) ;
D A and consequently the squares of E A and AF will be double the squares of EG and G A (Theo. 50.); or as the square of G A is equal to the squares of A C and CG (Theo. 44.), the
of EA and AF will be equal to double the squares of EG, GC, and A C. But the squares of E G and GC are equal to the square of E C, (Theo. 44.) or of its equal BC, Hence the squares of EA and AF are equal to double the squares of B C and AC; or as the squares of A B and AD are also equal to double the squares of B C and AC, (Theo. 50.) the squares of E A and AF are equal to the squares of B A and A D.
THEOREM LXIX. If two lines A C, DF, intersect each other at right angles at any point B in a circle, the sum of the squares of the four segments A B, B C, BD, and B E, are together equal to the square of the diameter. Draw CF parallel to DE, and join A E, AF, EF,
D and DC. Then as CF is parallel to D Е, and the angle ABE is a right angle, the inward and op- А.
B posite angle A CF is also a right angle, (Theo. 19.) and therefore ADCF is a semicircle, and A F a
(Theo. 53. Cor. 3.) Again as D E is parallel to CF, EF, and DC are equal. But the square of DC, or the square of E F, is equal to the squares of D B and BC (Theo. 44.); and the square of A E is equal to the squares of A B and B E. Hence the squares of A E and EF are equal to the squares of A B, BC, BD, and BE. But the square of A F is equal to the squares of AF and E F, therefore the square of A F is equal to the squares of A B, BC, BD, and B E.
OF PROPORTION. PROPORTION is the ratio, or numerical relation, which one quantity bears to another.
Quantities between which proportion can exist, must be of the same kind, as a line and a line, a surface and a surface, a solid and a solid.
A greater quantity is said to be a multiple of a less, when it contains the less a certain number of times, without any remainder ; and quantities so related, are said to have the same ratio or proportion to each other, that unity, representing the less, has to as many such units as are contained in the greater.
If a quantity, as A, be contained exactly a cer- A tain number of times in another quantity, B, the B quantity A is said to measure the quantity B; and if the same quantity. A be contained exactly a certain number of times in another quantity C; A is also said to be a measure of the quantity C; and F it is called a common measure of the quantities B and C; and the quantities B and C will, evidently, bear the same relation to each other, that the numbers do which represent the multiple that each quantity is of the common measure A.
Thus if B contain A three times, and C contain A also three times, B and C being equimultiples of the quantity A, will be equal to each other. And if B contain A three times, and C contain A four times, the proportion between B and C will be the same as the proportion between the numbers 3 and 4.
Again, if a quantity, D, be contained as often in another quantity, E, as A is contained in B, and as often in another quantity, F, as A is contained in C, the ratio of E to F, or the proportion between them, will be the same as the proportion between B and C, and in that case the quantities, B, C, E, and F, are said to be proportional quantities ; a relation which is commonly expressed thus, B :C::E:F.
Hence, in finding the proportion between two quantities, it is necessary to refer to a quantity which is conceived to measure them both, and the method of finding such a common measure, may be thus explained by an example.
Suppose it were required to find a common
A measure of the quantities A B and C D. From the greater, C D, take the less, A B, as often as it is contained in it; and from A B take
É D the remainder, E D, as often as it is contained in it; and from ED, the first remainder, take FB, the second, as often as it is contained in it; and so on, till nothing remain ; then the last remainder will be a common measure both of the original quantities, A B and C D, and of the several successive remainders. For as the last remainder measures the preceding one, and is also a measure of itself, it is a measure of the several parts of which the next preceding remainder is composed ; and it is therefore a measure both of each preceding remainder, and of the original quantities A B and C D.
Let us, as an instance, suppose that D contains A B thrice, with the remainder DE; and that A B contains D E twice, with the remainder B F; and that E D contains B F twice exactly. Then A F, which is twice E D, contains B F four times; and therefore A B contains B F five times. Whence CE, which contains A B three times, contains B F fifteen times ; and as E D contains B F twice, the whole C D contains B F seventeen times; and B F is therefore a common measure of A B and C D.
The quantities to be compared may, however, be so related to each other, that the subdivision of the remainders, in the manner above described, will never terminate. But even in that case the operation may be continued till a quantity so small be obtained, that on its being applied as a measure of the two quantities, the remainders will be less than any quantities that can be assigned ; and hence the difference of the proportion between the two given quantities, and the proportion between the numbers which represent the nearest multiple that each of them is of this approximate common measure will be indefinitely small; and the proportions may therefore be considered as identical.
Like multiples of any two quantities, A B and C D, have the same proportion as the quantities themselves. Let A B be to C D as any number
A a 7 B (say 3) to any other number, as 4, or let A B contain three such equal parts as C c d e D those of which CD contains four. Let E f, fg, g F, be any like multiples of E 9 F A a, ab, b B, G h, h i, i k, k H, the same
G h i k H multiples of C c, cd, d e, then the
whole of G H of the whole of C D, that each part of the one is of the corresponding part of the other. And as the parts A a, ab, 6 B, C c, cd, &c. are all equal, their like multiples, E f,fg, Gh, &c. will also be equal.
Hence E F is to GH as 3 is to 4, which is the same proportion that A B has to C D. In the same way may the ty be proved, whatever numerical relation A B may have to C D.
Cor. As A B and C D are like parts of E F and G H, like parts of quantities have the same proportion as the wholes.
THEOREM LXXI. In any four quantities, A B, BC, DE, and EF, if AB:BC :: DE:E F, then A B:DE::BC:EF. If A B be to B C, or D E to E F, as
въ с any number, 4, to any number, 3; then A B will contain four such equal parts,
E e F A a, as those of which B C contains three, Bb; and D E will contain four such equal parts, D d, as those • of which EF contains three, Ee. Hence AB will be to D E as Aa to D d (Theo.70. Cor.); and B C also will be to E F as B b to E e, or as A a to D d, and consequently A B:DE::BC: EF.
Cor. If A B :BC ::DE: E F, then BC: A B :: EF:D E.
THEOREM LXXII. In any four quantities, A B, BC, DE, EF (see the last figure), if A B : BC :: DE : EF, then A B + BC: A B or BC::DE + EF:D E or E F.
For AB + B C, or A C, is to A B or BC, as the number of parts in A C is to the number of equal parts in A B or BC; or as the number of parts in D F is to the number of equal parts in D E or EF, that is as D F to D E or E F. In the same way it may be shewn that A B - BC: A B or BC::DE - EF:D E or E F.
THEOREM LXXIII. Triangles, as A B C D E F, between the same parallels, A E, C F, or that have the same altitudes, are to each other as their bases, A B and DE. For let A B be to D E as any number, 3, C
F for example, to any other number, as 4; that is, let A B contain three such equal parts, A a, ab, b B, whereof D E contains four, Dc, ce, ef, fE; and join C a, Cb, A a 7 B Dcef E Fc, F e, and F f. Then the triangles C A a, Cab, CbB, F D c, Fce, &c. are all (Cor. 1. Theo. 33.) equal ; and therefore the triangle A B C contains three such equal parts, as those of which DF E contains four. Hence the triangle A B C is to the triangle D F E' as 3 is to 4, which is the same proportion that their bases have.
Cor. Parallelograms and rectangles between the same parallels, or that have the same altitudes, are to each other as their bases ; for parallelograms are double of their respective triangles.
THEOREM LXXIV. If two triangles, A B C D E F, stand on equal bases, A B, DE, the triangles are to each other as the perpendiculars, C H, FI, drawn from their vertices to their bases. Let B P be perpendicular to A B and
P equal to C H, and let BQ be equal to
F FI. Then the triangle A B P is equal to the triangle A B C, and the triangle A B Q to the triangle D FE. But A BP
A H B D : ABQ:: BP:P Q (Theo. 73.); or A B : CDFE:: CH: FI.
Cor. 1. Parallelograms on equal bases are to each other as their altitudes.
Cor. 2. Rectangles on equal bases are to each other as their altitudes.
D means, B and C, or A.D = B.C.
For A.D:B.D:: A:B, or C :D; and C:D::B.C:B.D (Theo. 73. Cor.) ; whence A.D:B.D:B.C:B.D; therefore as A.B and B.C have the same proportion to B.D, A.D and B.C are equal to each other.
Cor. 1. Since, by the rules of proportion in numbers, when four quantities are proportional, the product of the extremes is equal to the product of the means; and, by this theorem, the rectangle of the extremes is equal to the rectangle of the means; it follows, that the area or space of a rectangle may be represented or expressed by the product of its length multiplied by its breadth. And a square may be represented by the measure of its side multiplied by itself. So that what can be shewn of such products is true also of squares and rectangles.
Cor. 2. When the two means are equal, their rectangle becomes a square ; and hence it follows, that when three lines are proportionals, the rectangle of the extremes is equal to the square of the
THEOREM LXXVI. If of four lines, A, B, C, D (see the last figure), the rectangle of A and D be equal to the rectangle of B and C, then will A, B, C, and D,