THEOREM XVII.

In a right angled triangle, if a perpendicular A D be drawn from the right angle B A C to the base BC, the triangles on each side of it are similar to the whole triangle, and to one another.

BECAUSE the angles A D B and B A C are both right angles; the angle A B D is common to the triangles B A C and A B D, which therefore have the remaining angles B A D and B C A equal, (Theor. 6.) and the triangles B A C and A BD are equiangular.—In like manner it may be demonstrated the triangle A CD is similar to the triangle B A C, and consequently the triangles A B D, A D C, and BAC are similar.

Cor.-Hence B D:AD::AD: D C, or the perpendicular A D is a mean proportional to the segments upon the base BD and DC; and B DxD C=A D. (Theor. 15. Cor. 1.)

THEOREM XVIII.

If two chords A B and C D intersect each other within a circle or without it, by being produced, the rectangle under the segments made by their intersection, and terminated by the circumference, are equal; that is, BE

EA=DE×EC.

JOIN BC and DA; because the angle DE A and B E C are equal, (Theor. 2.) and the angle C B A and C D A are also equal, the triangle B E C is similar to the triangle DE A; (Theor. 5. Cor. 2.) therefore AE:CE::DE: BE, or BEXA E-DEXCE. (Theor. 15.)

LOGARITHMS.

LOGARITHMS are a set of artificial numbers, arranged in tables, peculiarly adapted to facilitate the computation of natural numbers. Their properties are such, that the sum of the Logarithms corresponding to any two or more natural numbers, answers to the Logarithm of their product. The Logarithm of every number is expressed by one of the indices 0, 1, 2, 3, 4, 5, &c. with decimals annexed to each index; as the Log. of 1 is 0.0000000, of 10 is 1.0000000, of 100 is 2.0000000, of 1000 is 3.0000000, &c. Hence it appears, that the indices of the Logarithms form a series in Arithmetical progression, or have a common difference in each term, and the natural numbers answering to this series are in Geometrical progression, or every term is a certain multiple of the preceding one. In the following system of Logarithms, the numbers corresponding to every different integer of the indices are each a power of 10; for example, the Log. index 2 answers to 100=10%, and 3 answers to 1000=10%, &c. but if this series was changed by assuming the power of any other number corresponding to the same indices, another system of Log. would be formed, having the same peculiar properties as the above: for instance, if we had assumed the Geometrical series of numbers to be the powers of 8, then every different index of the Log. would correspond to a different powcr of 8; as the Log. index 1 would in this system answer to 8, and the

index 2 to 64-8%, and 3 to 512-83, &c. so that as any number may be taken in place of 10, there may be an infinite number of different systems of Logarithms.

Since in the following system O is the Logarithm of 1, 1 of 10, 2 of 100, and 3. of 1000, the Logarithms of the numbers lying between 1 and 10 must be each greater than 0, and less than 1, therefore the Logarithm of these are expressed by decimals of 1; as,

The same may be shown of the Logarithms of numbers between 10 and 100, or between 100 and 1000; for those between 10 and 100 must be each greater than 1 and less than 2, and between 100 and. 1000 each is greater than 2 and less than 3; as,

Note. To those who wish to understand thoroughly the construction of Logarithms,

I recommend for their perusal the Introduction to Hutton's Logarithmic Tables.

F

EXPLANATION OF TABLE 1.

THE first or left-hand column of the first page, which is marked N. contains the natural numbers from 1 to 100, and in the second column, marked L. are the corresponding Logarithms to each number.

The second, and all the remaining pages of this Table, are divided into eleven columns, the first of which is, as before, marked N. containing all the Numbers from 100 to 1000, and the remaining ten, marked 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, contain the corresponding Logarithms of all the numbers from 100 to 10,000.

The indices are not printed with any of the Logarithms, except those Numbers from 1 to 100, but the index of each must always be prefixed when used; it being understood, that the index 2 is to be put before the Tabular Logarithm for all numbers from 100 to 1000, and 3 from 1000 to 10,000, and so on, making the index always one less than the number of integer figures in the Number for which the Logarithm is taken. Also in the same columns the first figure after the index being the same for several lines, is not repeated except in column 0, when it changes, but which, likewise, must be always prefixed to the other figures-thus making every Logarithm of this Table to contain seven places besides the index.

There being a difference between the Logarithm of every different number, the parts of these differences are proportioned for every every integer from 1 to 10: they are arranged under the Logarithms of each page, to which they respectively belong, and titled PROPORTIONAL· PARTS. This Table has a column N. containing the Numbers, and another D. of the corresponding Differences; but the particular use of it will be better explained in the following examples.