equal to

'(8—a) (8—b) (8—c). In like manner, by dividing

8

the expression for the area successively by s-a, s—b, s—c, we find, according to the seventh corollary, that the radii of the circles touching a, b, c, externally are respectively

By taking the continual product of these four expressions, and contracting the result, we get s (s — a) (s — b) (s —c), which is equal to the square of the area: and hence, by expressing this in words, we have the following remarkable theorem: The continual product of the radi of the four circles, each of which touches the three sides of a triangle, or their prolongations, is equal to the second power of the area.

PROP. XXIX.-This important problem was proposed by Mr. Townley, and solved by Mr. John Collins, in the Philosophical Transactions for 1671. It is of much use in the surveying of coasts and harbours.

APPENDIX, BOOK IV.

This book is supplementary to the eleventh and twelfth of Euclid, containing a few additional propositions in solid geometry, and exhibiting, in particular, the method of computing the contents of pyramids, cones, and polyhedrons. The integral calculus affords the best and easiest means of investigating most of what remains of the mensuration of the surfaces and volumes of solids.

PROP. IV. COR. 2.-An easy method of computing the content of a truncated pyramid or cone, that is, the frustum which remains when a part is cut from the top by a plane parallel to the base may be thus investigated by the help of algebra. The solid cut off is (APP. IV. 3, schol. 2) similar to the whole; and therefore the areas of their bases will be proportional to the squares of their corresponding dimensions, and consequently to the squares of their altitudes. Hence putting V to denote the volume or content of the frustum, H and B the altitude and base of the whole solid, and h and b those of the solid cut off, if we put qH2 to denote B, since B: b :: H2 : h2, or B : b : : qH2 : qh2, we shall have (V. 14) b=gh2; and therefore (APP. V. 4, cor. 2) the contents of the whole cone and the part cut off are equal respectively to 39H3. and qh3; wherefore V=}q(H3—h3), or, by resolving the second member into factors, V=q(H2+Hh+h2) (H—h)=}(qH2+qHh+ gh2) (H-h). Now qH2 is equal to B, qh2 to b, qHh to a mean proportional between them, and H-h to the height of the frustum. Hence, to find the content of a truncated pyramid or cone, add together the areas of its two bases and a mean proportional

between them, multiply the sum by the height of the frustum, and divide the product by 3.

This admits of convenient modifications in particular cases. Thus, if the bases be squares of which S and s are sides, and if a be the altitude of the frustum, we shall have

V=ja (S2+Ss+s2)=}a (3Ss+S2—2 Ss+s2);

or, V-ja (3 Ss+(S—s)}=a (Ss+}(S—8)2}.

Hence, to find the content of the frustum of a square pyramid, to the rectangle under the sides of its bases, add a third of the square of their difference, and multiply the sum by the height. It would be shown in like manner (APP. I. 39, cor. 2) that if R and r be the radii of the bases of the frustum of a cone, and a its altitude,

VarRr+(R-r)}.

EXERCISES.

1. THE least straight line that can be drawn to another straight line from a point without it, is the perpendicular to it: of others, that which is nearer to the perpendicular is less than one more remote; and only two equal straight lines can be drawn, one on each side of the perpendicular.

2. Of the triangles formed by drawing straight lines from a point within a parallelogram to the several angles, each pair that have opposite sides of the parallelogram as bases, are half of it. 3. If, in proceeding round an equilateral triangle, a square, or any regular polygon, in the same direction, points be taken on the sides, or the sides produced, at equal distances from the several angles, a similar rectilineal figure will be formed by joining each point of section with those on each side of it.

4. If the three sides of one triangle be perpendicular to the three sides of another, each to each, the triangles are equiangular.

5. A trapezoid, that is, a trapezium having two of its sides parallel, is equal to a triangle which has its base equal to the sum of the parallel sides, and its altitude equal to their perpendicular distance.

6. Given the segments into which the line bisecting the vertical angle, divides the base, and the difference of the angles at the base; to construct the triangle, and compute the sides.

7. Within or without a triangle, to draw a straight line parallel to the base, such that it may be equal to the parts of the other sides, or of their continuations, between it and the base.

8. Given the perpendicular of a triangle, the difference of the segments into which it divides the base, and the difference of the angles at the base; to construct the triangle.

9. In the figure for the 47th proposition of the first book of Euclid, prove that CF is perpendicular to AD.

10. The angle made by two chords of a circle, or by their continuations, is equal to an angle at the circumference standing on an arc equal to the sum of the arcs intercepted between the chords, if the point of intersection be within the circle, or to their difference, if it be without: (2) the angle made by a tangent and a line

cutting the circle, is equal to an angle at the circumference on an arc equal to the difference of the arcs intercepted between the point of contact and the other line: and (3) the angle made by two tangents is equal to an angle at the circumference standing on an arc equal to the difference of those into which the circumference is dívided at the points of contact.

Cor. If a tangent be parallel to a chord, the arcs between the point of contact, and the extremities of the chord are equal.

11. Given the sum of the perimeter and diagonal of a square; to construct it.

12. On a given straight line describe a square, and on the side opposite to the given line, describe equilateral triangles lying in opposite directions; circles described through the extremities of the given line, and through the vertices of these triangles are equal.

13. To inscribe an equilateral triangle in a given square.

14. Given the angles and two opposite sides of a trapezium to construct it.

15. In a given circle to place two chords of given lengths, and inclined at a given angle.

16. In the second figure in page 83, if AM, BM be joined, the angle AMB is half of ACB.

17. If, in proceeding in the same direction round any triangle, as in Exercise 3, points be taken at distances from the several angles, each equal to a third of the side, the triangle formed by joining the points of section is one third of the entire triangle.

18. To describe a square having two of its angular points on the circumference of a given circle, and the other two on two given straight lines drawn through the centre. Show that there may be eight such squares.

19. Given the vertical angle of a triangle, and the segments into which the base is divided at the point of contact of the inscribed circle; to describe the triangle, and compute the sides.

20. If any three angles of an equilateral pentagon be equal, all its angles are equal.

21. Given two sides of a triangle, and the difference of the segments into which the third side is divided by the perpendicular from the opposite angle; to construct the triangle.

22. Given the vertical angle of a triangle, the line bisecting it, and the perpendicular; to construct the triangle.

23. From a given centre to describe a circle, from which a straight line, given in position, will cut off a segment containing an angle equal to a given angle.

24. In a given triangle to inscribe a semicircle having its centre in one of the sides.

25. Through three given points to draw three parallels, two of which may be equally distant from the one between them.

26. Given an angle of a triangle, and the radii of the circles touching the sides of the triangles into which the straight line bi. secting the given angle divides the triangle; to construct it. 27. In a rhombus to inscribe a square.

28. Given the lengths of the two parallel sides of a trapezoid, and the lengths of the other sides; to construct it.

29. Given one of the angles at the base, and the segments into which the base is divided at the point of contact of the inscribed circle; to describe the triangles.

30. To draw a tangent to a given circle, such that the part of it intercepted between the continuations of two given diameters may be equal to a given straight line.

31. To draw a tangent to a given circle, such that the part of it between two given tangents to the circle may be equal to a given straight line.

32. Given the vertical angle of a triangle, the difference of the sides, and the difference of the segments into which the line bisecting the vertical angle divides the base; to construct the triangle.

33. Given any three of the circles mentioned in the scholium to the fourth proposition of the fourth book of Euclid; to describe the triangle.

34. A straight line and a point being given in position, it is required to draw through the point two straight lines inclined at a given angle, and enclosing with the given line a space of given magnitude.

35. From two given straight lines to cut off equal parts, each of which will be a mean proportional between the remainders.

36. A square is to a regular octagon described on one of its sides, as 1 to 2 (1/2).

37. In a given triangle to inscribe a parallelogram of a given

area.

38. In a given circle to inscribe a parallelogram of a given area. 39. Through a given point between the lines forming a given angle, to draw a straight line cutting off the least possible triangle. 40. To divide a given straight line into two parts, such that the square of one of them may be double of the square of the other, or may be in any given ratio to it.

41. To produce a given straight line, so that the square of the whole line thus produced, may be double of the square of the part added, or in any given ratio to it.

42. Given the area of a right-angled triangle, and the sum of the legs; to construct it.

43. Given the area and the difference of the legs of a rightangled triangle; to construct it.

44. Given one leg of a right-angled triangle, and the remote segment of the hypotenuse, made by a perpendicular from the right angle; to construct the triangle.

45. Given the base of a triangle, the vertical angle, and the side of the inscribed square standing on the base; to describe the triangle.

46. Given the base of a triangle, and the radii of the inscribed and circumscribed circles; to construct the triangle.

47. To draw a chord in a circle which will be equal to one of the segments of the diameter that bisects it.

48. In a given circle to draw a chord which will be equal to the