9. Ratio is the relation of two quantities of the same kind with respect to quantity, or the fraction which is equal to the quotient of one divided by the other, is called their ratio; hence a ratio can only subsist between quantities of the same kind.

The ratio of one quantity with another is expressed by placing two points vertically between them, or by writing the quantity compared above a line, and that with which it is compared below it: thus, the ratio of 3 to 4 is expressed either by 3: 4, or ; the first term is called the antecedent, and the last the consequent.

A ratio, is called a ratio of equality, majority or minority, according as the antecedent is equal to, greater, or less than the consequent: thus, 3:3 is a ratio of equality; 5:3 a ratio of majority; and 3: 5 a ratio of minority. When the terms of a ratio are multiplied or divided by the same number, the ratio itself is not altered.

Ratios may be compared by reducing the fractions which represent them to a common denominator: thus, 34 is a less ratio than 4: 5, because is less than, or 15 less than 18 by

2

Ratios of minority increase, and ratios of majority decrease, by adding the same quantity to each term: thus, if to 35 there be added 3, the resulting ratio is 6: 8, or 3: 4, which is greater than 3:5. Also, if to the ratio 53, there be 2 added, the resulting ratio is 7:5, which is less than 5:3.

Ratios are compounded by multiplying their antecedents together, and their consequents together: thus, the ratio of 35, compounded with the ratio of 4: 3, is 3 X 4:3 X 5, or 12 15, or 4: 5.

10. Proportion, is the comparison of two ratios.

11. Progression, is either Arithmetical or Geometrical. Arithmetical Progression is when three or more numbers increase or decrease by a constant number, as 1,3,5,7, &c. which is an increasing progression, the common difference of which is 2. Also, 20, 17, 14, 11, &c. is a decreasing progression, the common difference of which is 3. Whence Arithmetical Progression is formed by addition or subtraction.

12. Geometrical Progression is when three or more numbers have successively the same ratio, as 1, 2, 4, 8, &c. which is an increasing progression, its ratio or common multiplyer being 2. Also, 16, 8, 4, 2, &c. is a decreasing progression, the ratio of which is, whence Geometrical Progression is formed by multiplication or division.

13. Harmonical Proportion, is when the first term is to the third as the difference between the first and second is to the difference between the second and third; or in four terms, when the first is to the fourth as the difference between the first and second is to the difference between the third and fourth; or the reciprocals of numbers in Arithmetical Proportion are in Harmonical Proportion. MULTIPLICATION.

2

3

2 3 | 4 | 5 | 6 | 7 8 | 9 | 10 | 11 | 12
468 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24

1

6 9 1215|18|21| 24 | 27 | 30 | 33 | 36 4|8|12|16| 20 | 24 | 28 | 32 | 36 | 40 | 44 | 48 5 10 15 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 |60| 6 | 12 | 18|24| 30 | 36 | 42 | 48 | 54 | 60 | 66 | 72 14|21|28|35 | 42 | 49 | 56 | 63 | 70 | 77 |84 8 16|24|32| 40 | 48 | 56 | 64 | 72 | 80 | 88 |96 9 |18|27|36|45 | 54 |63|72 | 81 | 90 | 99 |108

7

10|20|30|40|50|60|70|80|90|100|110|120| 11|22|33| 44 | 55 | 66 | 77 | 88 | 99 |110|121|132| 12|24|36|48|60|72 | 84 | 96 |108|120|132|144

960 = 240 201 Pound.

N. B. A Moidore, 27s.-A Guinea, 21s.-A Mark, 13s. 4d.-An Angel, 6s. 8d.-A Crown, 5s.-Half-Crown, 2s. 6d.

By this weight are weighed Gold, Silver, and Jewels only.

N. B. The Standard for Gold Coin is 22 Carats of fine Gold, and 2 Carats of Copper, melted together. For Silver, it is 11oz. 2dwts. of fine Silver, and 18dwts. of Copper. AVOIRDUPOIS WEIGHT.

There are several other denominations in this Weight, that are used in some particular Goods, viz.

The Apothecaries mix their medicines by this weight; but they buy and sell by Avoirdupois weight.

The pound and ounce in this weight are the same as those in Troy Weight, but the smaller divisions are different.

By this measure, all things that have length and breadth are measured; as land, paving, plastering, roofing, tiling, flooring, glazing, &c. Land is measured by a chain, called Gunter's Chain, which is 4 poles or 22 yards long; and consists of 100 links. Ten chains in length, and one in breadth make an acre.

NOTE. 49 square yards make a rood of digging.
CUBIC, OR SOLID MEASURE.

By this, all things that have length, breadth, and thickness, are measured.

This is on a great circle of the earth; that is, a circle that divides the earth into two equal parts. Every circle contains 360° degrees, each degree, 60' minutes, each minute, 60" seconds, &c.