An introduction to linear drawing/Chapter 10
PROBLEMS
in
Arithmetick and Geometry.
It is easy to unite two branches of instruction, which are so important and so analogous. 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 hun- dredths, and two thousandths of an inch ; but the for- mer way is preferable.*
In Addition and Subtraction, columns of the same name should be placed under each other, and the cal- culation 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 7,08 549,391 Add the following sums : 36,075 8,1 4,44 9,6 28,04 8,176 345,56 686,008 . 0,43 86,115 5,16 10,08 6,8 82,686 2,5
Subtract the following : 68,06 4,85 15,908
17,67 3,9 12,819
The master or monitor may vary such sums at pleasure.
- After thousandths, come ten thousandths, hundred thousandths, mil-
Konths, &c. but for ordinary uses we seldom come down to so small a fraction. 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:
4,37 183,2
2,3 0,24
1311 7328
874 3664
10,061 43,968
Multiply 37,04 by 4,8
3,96 by 0,84
18,5 by 5,18
468,007 by 8,14
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,u place a comma after them, and then find the decimals by putting a cypher 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. examples.
To divide 10,051 by 4,37 I write thus:
4370) 10051 (2,3 8740
13110 13110
0
That is, I add a cipher to 4,37 hundredths, to make them thousandths, because there are thousandths in the dividend.
In the above sum, the answer is, 2 wholes and 3 tenths.
Again, divide 154,3 by 21,26.
2126) 15430 (7,25 14882
5480 4252
12280 10630
1650 remainder.
The answer is 7 wholes, and 25 hundredths. 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