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The cuts of the three first classes are copied from the French original; those of the fourth and fifth classes, from drawings by pupils of the Monitorial School. The copperplate was designed by the translator. For the purpose of exhibiting the four principal orders, and affording an opportunity of comparing their relative height, grace, and strength, the module of each order is the same, viz. three twentieths of an inch.

PROBLEMS

IN

Arithmetick and Geometry.

2000

It is easy to unite two branches of instruction, which are so important and so analagous. Artists and mechanicks ought to be able themselves to measure their work, whatever it may be; and to draw plans, to make contracts for work, to calculate the price, and quantity of materials necessary for the work; and in fine, to make all the estimates required by the art they practise.

To enable them to do this, we shall unite the elements of Geometry and Arithmetick, explain the problems and rules of most common occurrence, and add numerical examples to illustrate their application. The master will vary the examples at pleasure.

Inches are divided into tenths, hundredths, thousandths, &c. and calling the inch unity, or a whole, we place a comma at the right hand of it to separate the fractions or parts. For example, to express 8 inches and 6 tenths, we write 8,6; for 9 inches and 72 hundredths, we write 9,72 ; for 10 inches and 626 thousandths, we write 10,626, and so on. If there be no whole inches, a cipher is put in the place of inches, and the comma as before, thus, 0,382 stands for 382 thousandths of an inch, or as the first column at the right of

the comma is tenths, the second hundredths, and the. third thousandths, we may read it, 3 tenths, 8 hundredths, and two thousandths of an inch ; but the former way is preferable.*

In ADDITION and SUBTRACTION, columns of the same name should be placed under each other, and the calculation made as if there were no decimal fractions. The following examples will show the use of this rule. Addition.

Subtraction. 432,178

324,15

30,4 17,231

187,3

19,28
9,4
83,502

136,85

11,12 7,08

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The master or monitor may vary such sums at pleasure.

* After thousandths, come ten thousandths, hundred thousandths, millionths, &c. but for ordinary uses we seldom come down to 50. small a frac:ion:

MULTIPLICATION is performed as if there were no comma ; but in the sum total, as many figures must be cut off at the right hand of the comma, as are cut off in both the multiplier and sum multiplied. For example:

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In Division, add ciphers at the right of whichever of the two numbers has the least number of decimal figures, that the dividend or sum to be divided, and the divisor or sum to divide by, may have an equal number of them; then pay no regard to the comma, and divide as in ordinary arithmetick.

When you have found the wholes of the quotient, place a comma after them, and then find the decimals by putting a cipher at the right of the remainder, and dividing anew, you will then have the first figure after the comma in the quotient. Add another cipher to the remainder, and you will have another decimal figure, and so on.

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To divide 10,051 by 4,37 I write thus :

4370) 10051 (2,3

8740

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That is, I add a cipher to 4,37 thousandths, to make them 10 thousandths, because there are 10 thousandths in the dividend. 10 thousandths divided by 10 thousandths, will give 10 thousandths for the quotient ; hundredths divided by hundredths, give hundredths, &c.

In the above sum, the answer is, 2 ten thousandths and 3 tenths of a ten thousandth.

Again, divide 154,3 by. 21,26.
21,26) 15430 (7,25

14882

5480
4252

12280
10630

1650 remainder.

The answer is 7 hundredths, and 25 hundredths of a hundred. It is unnecessary to carry the remainder to any lower fraction.

Divide 36,75 by : 8,4

460,8 by 46,54

84,968 by 8,68 166,14 by 19,762 86,4 by 4,86

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