PART SECOND.

THE SQUARE ROOT OF TWO,

OR

THE COMMON MEASURE OF THE SIDE AND DIAGONAL OF THE SQUARE;

BEING A

SHORT, EASY, AND CONVENIENT METHOD OF FINDING EITHER THE SIDE OR DIAGONAL OF THE SQUARE, WHEN THE OTHER IS KNOWN, BY

COMMON MULTIPLICATION AND DIVISION;

ALSO,

THE SQUARE ROOT OF TWO, BY DIVISION ALONE, TO ONE HUNDRED AND FORTY-FOUR DECIMAL PLACES.

(135)

THE SQUARE ROOT OF TWO.

THE following pages are intended to explain the nature, uses, and advantages of the Common Measure, as applied to Civil Engineering, Architecture, Draughting, Machinery, Building, Painting, and Landscape Gardening.

Numerous instances could be given where it has been already tested and found to be everything that could be desired.

According to PROPOSITION 1, COR. PART I, the side of any square is to the dioganal as one is to the square root of two; that is, if the side of any square be ONE (1), the diagonal must be the square root of two (2); therefore, by the ordinary method, to find the diagonal of a square, when the side is known, use the following

RULE.

Take as many figures of the square root of two as will make the result sufficiently exact, and multiply these figures by the number of units in the side of the given square, the product will be the true diagonal in terms of the side of the square.

NOTE. If the side of the square be expressed in inches, the diagonal will also be expressed in inches; but if the side be expressed in feet or rods, etc., the diagonal will be expressed in feet or rods, etc.; and if the side contain a decimal or a fraction, the result of that decimal or fraetion will have its corresponding result in the diagonal. In all cases the requisite number of figures of the square root of two must first be obtained.

When the diagonal is given, to find the side, it is not so easy a task, and requires more time and labor than to find the diagonal when the side is known; but, for the benefit of those who still prefer the former method, I shall here insert the rule which embraces two cases.

CASE 1.

If the diagonal is expressed in integers, such as 1, 2, 3, etc., the side of the square is found by dividing these numbers by the square root of two; the result will be the side in the same denomination as the diagonal.

CASE 2.

If the diagonal contains either a fraction or a decimal, this fraction must first be reduced to a decimal and the value pointed off as in reduction of decimals. If a decimal only, it must first be pointed off, then divide this decimal by the square root of two, according to the rule for division of decimals, the quotient will be the side of the square in terms of the diagonal; but, in all cases, the requisite number of figures of the square root of two must first be found.

To find an exact common measure of the side and diagonal of the square would be equivalent to finding the exact value of the square root of two; but the square root of two is an irrational quantity, therefore it has no end. It is believed that the following common measure comes nearer to an exact common measure than any that has been heretofore found, as it can be extended ad infinitum, and when either the side or the diagonal of any square is known the other can be found by common multiplication and division.

Owing to the difficulty of working out the necessary figures of the square root of two, and the tax upon the mind necessary to remember the same, civil engineers, architects, draughtsmen, builders, etc., have long since felt the want of a COMMON MEASURE expressed in integers, or a series of numbers, which would express the relation between the side and diagonal of the square; one that could be easily remembered, plainly understood, and readily applied. Such, for example, as the ratio between the circumference and diameter of the circle, found 355 113'

by Motius in 1640, namely:, or 113|355, which it is said will give

the ratio to six decimal places correct, viz.: 3.141592.

The author is happy to state that such a common measure of the side and diagonal of the square has been found which may be expressed in integers, and he ventures to hope that when it is fully tested the old method of squaring the side, doubling and extracting the square root, will be gladly cast aside as a noble relict of "Auld Lang Syne," when simpler and easier methods were unknown.

The common measure here introduced combines all the advantages of conciseness, simplicity, and perfection, for by its aid the desired results are reached much more rapidly than by any former method and with equal exactness.

FIRST EXAMPLE.-Suppose a builder is laying the foundation for a house and wants to make each of the corners a right angle—that is, a perfectly square corner; how will he do it?

ANSWER. First measure 10.5 feet on each side of the right angle in the direction of the sides, commencing at the angular point; then the distance across from the two extreme points will be 14.85 feet, or 14 feet, 10.2 inches. This result is found by multiplying the side of the given square by 99 and dividing the product by 70, which gives the desired result. See Plate 8, Figure 1.

SECOND EXAMPLE.-Suppose a civil engineer, while surveying, wants to lay out a right angle without the use of his instruments; how will he do it?

ANSWER.-Measure with a tape line any distance, on either side of the right angle, say 105 feet, in the direction of the sides, commencing at the angular point. Then will the distance across from the extreme points be 148.5 feet, or 148 feet 6 inches. This result is found precisely like the former, namely, by multiplying the side of the given square by 99, and dividing the product by 70, which gives the desired result. See Plate 8, Figure 2.

THIRD EXAMPLE.-Suppose a mechanic has a given circle, the diameter of which is 9.9 feet, or 9 feet 10.8 inches, and he wants to find the side of the largest square which it is possible to inscribe in the given circle; how will he find it?

ANSWER.-Multiply the given diameter by 70, and divide the product by 99, which will give the desired result, which will be 7 feet. See Plate 8, Figure 3.

If any person has a desire to test the correctness of these results by the former method, he is at liberty to do so, and will find them correct to the fourth decimal place.

NOTE. In the third example above, the diameter of the circle may be regarded as the diagonal of the given square, the side of which was required to be found; therefore the square which is so formed within the given circle is said to be inscribed within it.

From these three examples, which embrace two cases, we deduce the following rules:

CASE 1.

When the side of any square is known, to find the diagonal correct to four places of figures.