PROBLEM XXVIII.

721. The three distances from an oak, growing in an open plain, to the three visible corners of a square field, lying at some distance, are known to be 78, 59.161, and 78 poles, in successive order. What are the dimensions of the field, and its area?

Ans. Side of the square 24 rd.; area 3 A. 2 R. 16 rd.

PROBLEM XXIX.

722. There is a house of three equal stories in height. Now a ladder being raised against it, at 20 feet distance from the foot of the building, reaches the top; whilst another ladder, 12 feet shorter, raised from the same point, reaches only to the top of the second story. What is the height of the building? Ans. 41.696 ft.

PROBLEM XXX.

723. The solidity of a cone is 2513.28 cubic inches, and the slant side of a frustum of it, whose solidity is 2474.01, is 19.5 inches. Required the dimensions of the cone.

Ans. Altitude 24 inches; base diameter 20 inches.

PROBLEM XXXI.

724. Within a rectangular garden containing just an acre of ground, I have a circular fountain, whose circumference is 40, 28, 52, and 60 yards distant from the four angles of the garden. From these dimensions, the length and breadth of the garden, and likewise the diameter of the fountain, are required.

Ans. Length 94.996 yds.; width 50.949 yds. ; diameter of the fountain 20 yds.

PROBLEM XXXII.

725. There is a vessel in the form of a frustum of a cone, standing on its lesser base, whose solidity is 8.67 feet, the depth 21 inches, its greater base diameter to that

of the lesser as 7 to 5, into which a globe had accidentally been put, whose solidity was 24 times the measure of its surface. Required the diameters of the vessel and of the globe, and how many gallons of water would be requisite just to cover the latter within the former.

Ans. 35 and 25 inches, top and bottom diameters of the frustum; 15 inches, diameter of the globe; and 34.2 gallons, the water required.

PROBLEM XXXIII.

726. Three trees, A, B, C, whose respective heights are 114, 110, and 98 feet, are standing on a horizontal plane, and the distance from A to B is 112, from B to C is 104, and from A to C is 120 feet. What is the distance from the top of each tree to a point in the plane which shall be equally distant from each? Ans. 126.634 ft.

PROBLEM XXXIV.

727. A person possessed a rectangular meadow, the fences of which had been destroyed, and the only mark left was an oak-tree in the east corner; he however recollected the following particulars of the dimensions. It had once been resolved to divide the meadow into two parts by a hedge running diagonally; and he recollected that a segment of the diagonal intercepted by a perpendicular from one of the corners was 16 chains, and the same perpendicular, produced 2 chains, met the other side of the meadow. Now the owner has bequeathed it to four grandchildren, whose shares are to be bounded by the diagonal and perpendicular produced. What is the area of the meadow, and what are the several shares?

Ans. Area of the whole meadow, 16 acres; shares, 1 R. 24 rd.; 1 A. 2 R. 16 rd.; 6 A. 1 R. 24 rd.; 7 A. 2 R. 16 rd.

ELEMENTS

OF

PLANE AND SPHERICAL

TRIGONOMETRY;

WITH

PRACTICAL APPLICATIONS.

It thus appeare that, in the common system, the logarithm of every number between 1 and 10 is some number between 0 and 1; that is, a proper fraction. The logarithm of every number between 10 and 100 is some number between 1 and 2; that is, 1 plus a fraction. The logarithm of every number between 100 and 1,000 is some number between 2 and 3; that is, 2 plus à fraction; and so on.

4. By means of negative exponents the application of logarithms may be extended, in the common system, to numbers less than 1. Thus, since

0.1;

0.01;

0.001;

0.0001;

&c.

From this it appears that the logarithm of every number between 1 and 0.1 is some number between 0 and -1; that is, -1 plus a fraction. The logarithm of every number between 0.1 and 0.01 is some number between 1 and -2; that is, -2 plus a fraction. The logarithm of every number between 0.01 and 0.001 is some number between -2 and -3; that is, -3 plus a fraction; and so on.

5. In the common system, as the logarithms of all numbers which are not exact powers of 10 are incommensurable with those numbers, their values can only be obtained approximately, and are expressed by decimals.

6. The integral part of any logarithm is called the CHARACTERISTIC, and the decimal part is sometimes called the MAN

TISSA.

7. The characteristic of the logarithm of ANY NUMBER GREATER THAN UNITY, is one less than the number of integral figures in the given number.

For it has been shown (Art. 3) that the logarithm of 1 is 0, of 10 is 1, of 100 is 2, of 1000 is 3, and so on.

8. The characteristic of the logarithm of ANY DECIMAL FRACTION is a negative number, and is one more than the number of ciphers between the decimal point and the first significant figure.

For it has been shown (Art. 4) that the logarithm of 0.1 is -1, of 0.01 is -2, of 0.001 is -3, and so on.

NOTE. In general, whether the given number be integral, fractional, or mixed, the characteristic of the logarithm of any number expressed decimally is the distance of the first, or left-hand, significant figure from the units' place, being positive when that figure is on the left of the units' place, and negative when on the right.

GENERAL PROPERTIES OF LOGARITHMS.

9. The logarithm of a PRODUCT is equal to the sum of the logarithms of its factors.