and the adjacent points in which it cuts the circle may be to one another; the intercepts of the produced parts between the parallelogram and the ruler shall be the required means.

For the rectangles under the whole produced sides and external intercepts are to one another, being each

to the rectangles under = segments of the ruler; .. the whole produced side opposite the first extreme: the other whole produced side :: the external segment of the latter the external segment of the former (14. 6.), but the whole produced sides are to one another :: first extreme the first intercept, and the produced sides are also to one another: the second intercept: the last extreme; .. the first extreme: the first intercept :: the second intercept the last extreme, .. those intercepts are the required means.

Des Cartes method.

For this method there is necessary an instrument formed of two rulers joined at an angle, and moveable on a pivot at its vertex, so as to close or open at pleasure ; and having perpendicular rulers alternately inserted between them, so that on opening the first rulers the perpendicular rulers may push one another forward. Then on the ruler in which the first perpendicular is inserted, lay off from the vertex the first of the given extremes, and on the other ruler lay off also from the vertex the last extreme; then fixing the first perpendicular ruler at the extremity of the first extreme, open the rulers till the third perpendicular passes through the extremity of the last extreme; then the intercepts of the rulers between the vertex and the second and third perpendiculars will be the required means; this is evident from Cor. Prop. 8.

By this last method any number of means may be found between two given extremes. By adjusting the rulers, so that the first perpendicular will be at the extremity of the first extreme, and then open the rulers till the perpendicular, whose number is one more than the number of required means, will pass through the extremity of the last extreme.

If the number of required means be odd, the extremes must be both laid off on the ruler in which the first perpendicular is inserted.

PROP. XIV. THEOR.

Equal parallelograms, which have one angle in each equal, have the sides about the equal angles reciprocally proportional. And if two parallelograms have one angle in each equal, and the sides about the equal angles reciprocally proportional, the parallelograms shall be equal.

(Note. Parallelograms which have an angle in each = are mutually equiangular (Cor. 2. 34. 1). By reciprocally proportional, is meant, that a side about the = angle of the first: a side about the angle of the second: the other side about the equal angle of the second : the other side about the angle of the first.)

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PART 1. Conceive the parallelograms to be so placed that the given = s may be vertically opposite; complete the parallelogram of which two sides that do not lie in directum are sides. Then the first parallelogram: this

of the first : a side کے = completed one :: a side about the

about the = 2 of the second, in directum with it (Prop. 1. 6); and the second par. : this completed one: the other side about : Z of second : the other side about: = L of the first, ... since the given paral. are =, a side (about the = ) of first par. : a side (about the other side (about the <<), of second: the other side (about the <) of first; .. the sides about the = 4s are reciprocally proportional.

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of the second : : the ( ے

PART 2. The same construction remaining. Since the reciprocal ratios of the sides are given, and since the given parallelograms severally bear these ratios to the constructed parallelogram, the given parallelograms bear the same ratio to the same parallelogram, and are there

Equal triangles which have one angle in each equal, have the sides about the equal angles reciprocally proportional. And two triangles, which have one angle in each equal, and the sides about the equal angles reciprocally proportional, are equal.

Conceive the given As so placed that the equals shall be vertically opposite, connect the extremities of two

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sides about thes, that do not lie in directum, then the demonstration of each part will be precisely as in the foregoing proposition, save that you substitute the word triangle for parallelogram.

Cor. 1. From this proposition, and the preceding, it is evident, that triangles and parallelograms whose bases and altitudes are reciprocally proportional, are equal.

Cor. 2. If two triangles have two sides reciprocally proportional to two, and the angles contained by those sides together equal to two right angles, the triangles are equal. For complete the parallelograms of which they are halves, and those parallelograms will be found mutually equiangular, and the sides about theirs reciprocally proportional, .. the parallelograms are =; and .. the As which are their halves are =.

Cor. 3. In like manner if two equal triangles have two angles together equal to two right angles, the sides about those angles shall be reciprocally proportional.

This proposition furnishes us with two cases of = As in addition to those of the first Book.

PROP. XVI. THEOR.

If four right lines be proportional, the rectangle under the extremes shall be equal to the rectangle under the means. And if the rectangle under the extremes is equal to the rectangle under the means, the four right lines are proportional.

PART 1. Draw right lines to the first and second; and from the extremity of that to the first, erect a perpendicular to the fourth, also from the extremity to the second erect a perpendicular third, and complete the rectangles.

of that

= to the

The base of the first rectangle: the base of the second :: the altitude of the second: the altitude of the first, and the rectangles are === (14. 6).. &c.

PART 2. The rectangles being given, their bases and altitudes are reciprocally proportional, but the base and altitude of one are to the extremes, and of the other to the means, .. &c.

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=

Cor. 1. If four right lines be proportional, any pa

rallelogram under the extremes is equal to an equiangular parallelogram under the means.

Cor. 2. In any triangle, the square of the line besecting any angle, internal or external, is equal to the difference between the rectangle under the sides about that angle, and the rectangle under the segments of the sides which the bisecting line meets.

Circumscribe a circle about the given A, produce the bisecting line to meet its periphery, and connect the extremity which meets the periphery with the extremity of either side about the bisected. Then the s at the periphery standing on the arch of which this side about the bisected is the chord, are = (21. 3), as are also the parts of the bisected; .. the As to which those s belong are similar, .. one of the sides containing the bisected the bisecting line: : the whole produced line: the other side containing the bisected ;.. the rectangle

:

under the sides is to the rectangle under the bisecting line and it produced, i. e. to the of bisecting_line and rect. under it and produced part, i. e. to the of bisecting line and rect. under the segts. of the side subtending bisected.

Cor. 3. If from any angle of a triangle, a perpendi cular be let fall on the opposite side, or opposite side produced, the rectangle under the sides containing that angle is equal to the rectangle under the perpendicular and the diameter of the circumscribing circle.

Draw a diameter from the vertex of the from which the perpendicular is let fall, and join its other extremity with either extremity of the side subtending that Then the A thus formed will be right angled, and will be similar to a ▲ of which the perpendicular is a side, and whose subtended by this perpendicular is at the other extremity of the side subtending the from which the perpendicular falls. Therefore one of the sides containing the the perpendicular:: the diameter: the other side containing the ;. &c.

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Cor. 4. In a quadrilateral figure inscribed in a circle, the rectangle under the diagonals is equal to the sum of the rectangles under the opposite sides.

From the vertex of any of the 4s, (if that be not bisected by the diagonal,) draw a right line cutting rom the greater part, and with the side of the given

figure, an to the less; that line will divide the diagonal on which it falls into segments, the rect. under either of which and the other diagonal, is to the rect. under the side of the (from the vertex of which the line was drawn), conterminous with that segt., and the opposite side of the given figure. Because that segt. and the side conterminous with it are sides of a A, similar to one of which the opposite side of the given figure and the other diagonal are sides, .. that segt. and diag. are extremes, and the opposite sides of the given figure are means; (those As are similar, having two s standing on the same arc, and two others = by construction.) Therefore the sum of the rectangles under one diagonal, and the segts. of the other, which sum is to the rectangle under the two diagonals (1. 2.), is to the sum of the rectangles under the opposite sides.

Note 1. It is on principles similar to those of this proposition, that ther ule of three is founded; in it we are given three terms of an analogy to find a fourth; if the given terms be the extremes, and a mean, we can find the other mean, for the mean sought will be the quotient arising from the division of the product of the given extremes by the given mean, ex. gr. let the given terms be ::8: 10 the product of 4X 10 = 40, and 40÷8 .. 5 is the mean sought; in like manner if we are given any other three terms, we can find the fourth.

4:

5

Note 2. Thus also we can apply to a given right line a rectangle = to a given one; by finding a fourth proportional to the given line, the base of the given rectangle and the altitude of the given rectangle; that fourth proportional will be the altitude of the required rectangle.

We can also (by Cor. 1.) apply to a given right line a parallelogram = and equiangular to any given parallelogram; for a fourth proportional to the given line, the side of the given parallelogram next in magnitude to it (the given line), and the other side of the given parallelogram will be the required side of the par. to be constructed, &c.

Note 3. By being given the altitude and side of an isosceles A, we can find the diameter of the circumscribing