Imágenes de páginas
PDF
EPUB

Draw a square EBGD (Fig. 51), having each of its sides equal to the sum of a+c, and a second square HFMN (Fig. 52) of the same size.

[blocks in formation]

The two squares which are shaded are the squares on the sides a and b.

[blocks in formation]

Along the four sides of the square EBGD mark off a length

equal to c from B to A, G to C, D to L, and E to P.

Join PL, LC, CA, and AP.

Then ACLP is an inscribed square, its side b units in length. The triangles 1, 2, 3, 4 in the square EBGD have their sides respectively equal to those marked in the same manner in the square HFMN, and the four triangles in the one square are equal in all respects to those in the other.

Hence, if we imagine these four triangles removed it follows that the inscribed square, its side b units in length, in the square EBGD is equal to the two squares in HFMN.

Thus in a right angled triangle, its perpendicular sides a and c and hypothenuse b,

b2= a2+c2.

From this relation when any two of the three sides are given, the third can be found.

Ex. 1. If the hypothenuse be 17 and the base 8, then

Since

height=√172-82 – 15.

172-82 (17+8) (178) = 25 × 9;

.. √25 x 9=5× 3 = 15.

By successive applications of a right-angled triangle the square roots of the first n natural numbers (1, 2, 3.....n) can be obtained.

[blocks in formation]

Ex. 2.

FIG. 53.

Draw a line AB=1 (Fig. 53), also BC perpendicular to

AB and equal to 1. Join AC, then AC-√2. Draw CD perpendicular to AC and equal to 1. Join CD, then AD=√√3.

Proceed in a similar manner to obtain the remaining values.

EXERCISES. XLI.

The following examples may be solved graphically:

1. The hypothenuse of a right-angled triangle is 177, and the base 141 find the remaining side.

2. If the two perpendicular sides of a right-angled triangle are 27 and 36, find the hypothenuse.

3. The base is 5-7 in., and the perpendicular 1 ft. 53 in. : find the hypothenuse.

4. One end of a ladder rests against a wall at a distance of 35 ft. from the ground; the bottom of the ladder is at a distance of 4 yards from the wall: find the length of the ladder.

5. Given the base of a right-angled triangle 12, and the perpendicular 16, find the length of the hypothenuse.

6. Given the base 182.7, and the hypothenuse 288, find the perpendicular.

7. Given the hypothenuse 560 14 ft., and the perpendicular 454 79 ft., find the base.

8. The sides of a rectangular field are respectively 563 and 369 links: find the diagonal.

9. A ladder 40 ft long just reaches to the top of a wall; the bottom of the ladder is 6 ft. from the wall: find the height of the wall.

10. Find the side of a square field whose area is 15 acres.

11. The perimeter of a right-angled triangle is 24 ft., and its base is 8 ft. find the other sides.

12. Find the hypothenuse of a right-angled triangle, one of its sides 39 in. and the area 84.5 sq. ft.

13. A field is in the form of a rectangle whose adjacent sides are 510 and 680 ft. respectively find the length of a path from one corner to the opposite one.

Summary.

Representative fraction of a scale

[ocr errors]

number of units in any line on drawing
the number of units the line represents

Right-angled triangles.—In a right-angled triangle the square on the hypothenuse, is equal to the sum of the squares on the other two sides.

CHAPTER X.

LABOUR-SAVING METHODS: LOGARITHMS.

MULTIPLICATION, DIVISION, INVOLUTION AND EVOLUTION BY LOGARITHMS.

The use of Logarithms to facilitate calculations.--On account of the labour involved in the processes of multiplication, division, involution and evolution, many labour-saving devices are used by practical men. By such means many operations which would otherwise require considerable labour, and consequent risk of error, are performed readily, and in many cases almost mechanically.

Amongst the many forms of labour-saving contrivances we can only refer to a few. Perhaps the most common is that of a carefully compiled table of numbers by means of which, knowing the weight or price of a single article, the weight or price of any number can at once be ascertained. This gives rise to the so-called "ready reckoners."

Also, in special cases where a large amount of multiplication, division, etc., has to be performed, one or other of the many forms of calculating machines may be used. These latter are far too expensive for general use. Hence, when it is necessary for any practical purpose to multiply or divide one set of numbers by another, the contracted methods shown on page 60 may be used. But multiplication and division as well as involution and evolution can be much more readily and easily performed by either logarithms or the slide-rule.

Logarithms and Logarithmic Tables.--As it is necessary that both the student and the practical man should be able to use logarithmic tables with ease and facility, and as they may

be used readily even by those who are not acquainted with the manner in which they are usually calculated and tabulated, it is desirable before entering into any explanation as to the principles on which common and hyperbolic logarithms are calculated to consider how, by means of a table of logarithms, the troublesome and tedious arithmetical operations of multiplication, division, involution and evolution are readily and accurately performed. We shall also find that many problems which would be impossible by arithmetical processes are readily solved when logarithms are used. Again, in using a slide-rule a knowledge of the use of logarithms is of service, enabling the significance of its divisions to be far more intelligible than would otherwise be the case.

Logarithms of numbers consist of an integral part called the index or characteristic, and a decimal part called the mantissa. If the reader will refer to Table IV., he will find that opposite any given number from 10 to 99 four figures are placed. These four figures are called the mantissa, the characteristic has to be supplied when writing down the logarithm of any given number in a way to be presently described.

Where great accuracy is required, seven or more figures are to be found in the mantissa. Logarithmic tables of all numbers from 1 to 100000 have been calculated with seven figures in the mantissa, but for all practical purposes, and where only approximate calculations are required, such a table as that shown in Table IV., and known as four figure logarithms, is very convenient. By means of the numbers 10 to 99, and (a) those at the top of the table, and (b) in the difference column on the right, the logarithm of any number from 0 to 10000 can be written down.

In logarithms all numbers are expressed by the powers of some number called the base.

DEF.-The logarithm of a number to a given base is the index showing the power to which that base must be raised to give the number.

Thus if N denote any number and a the base, then by raising a to some power x we can get N. This is expressed by the equation N=a*.

Any number could be used as the base, but, as we shall find,

« AnteriorContinuar »