When four quantities are proportional, the product of the numbers of the 1 st and 4 th terms, is equal to the product of the numbers of the 2 nd and 3 rd terms: thus, in the proportion This evidently arises from the nature of proportion; for as the ratio, 5-7 ths of the first set of terms, is equal to the ratio, 20-28 ths of the second set of terms, and as the latter terms are each 4 times the former, the products may be resolved into 5 × (7 X 4) = 7 × (5 × 4); and so for whatever may be the relation between the numbers of the first and third or the second and fourth terms. If the first term of a proportion is greater than the second, the third is greater than the fourth; if equal, equal; and if less, less. If the four terms of a proportion consist of similar quantities, the greatest and the least are together greater than the other two terms, as with the four terms, 5 s., 7 s., 20 s., and 28 s., which are in proportion, and of which the 5 s. and 28 s., are together greater than 7 s. and 20 s. When the relation of one quantity to another is the same as the relation compounded of several relations, they are then said to form a compound proportion. Thus, as the relation of 12 yards to 44 yards, is equal to the compounded relations of 7 pence to 9 pence, of 6 inches to 11 inches, and of 9 oz. to 14 oz., for the ratios, these quantities are said to form a compound proportion, and are expressed by characters, thus, 7 pence : 9 pence 6 inches : 11 inches :: 12 yards: 44 yards, 9 oz. : 14 oz. which are read, as 7 pence are to 9 pence, as 6 inches are to 11 inches, and as 9 oz. are to 14 oz., so are 12 yards to 44 yards. By separate proportions, the result is the same; for 7 pence : 9 pence: 12 yards: 15 yards* 11 inches :: 153 yards*: 28 yards+ 6 inches 9 oz. And the relation of 12 yards to 44 yards is said to be compounded of the relations of 12 yards to 15 yards, of 15 yards to 28 yards, and of 282 yards to 44 yards. PROGRESSION. Numbers or quantities are said to be in progression when they increase or decrease either by equal differences, or by the multiplication by the same number. When the successive terms of a series have equal differences, they are said to be in arithmetical progression; and when they are produced in succession by the operation of the same multiplier, they are said to be in geometrical progression. In all progressions five things are taken into consideration; viz. the first term, the last term, the number of the terms, the difference or ratio of the terms, and the sum of the terms; any three of which being given the other two may be found. ARITHMETICAL PROGRESSION. Arithmetical Progression is when three or more numbers or quantities increase or decrease by a common or equal difference; as 2, 5, 8, 11, or 12, 10, 8, 6, 4, 2. When the number of the terms is equal, the sum of the extremes is equal to the sum of the means; but when the number of the terms is unequal, the sum of the extremes is equal to twice the mean term. The sum of the extremes is also equal to the sum of any two terms at equal distance from the extremes. The number of the differences is equal to the number of the terms less 1. With these data we have the following Rules. To find the sum of the terms-Multiply the sum of the two extremes by half the number of the terms. To find the difference between the terms-Divide the difference between the first and last terms, by the number of the differences, that is, by the number of the terms less 1. To find the number of the terms—Divide the difference between the first and last terms by the common difference, and add 1 to the quotient. To find the last term-Multiply the number of the differences less one by the difference, and to the product add the first term when the series is increasing, or take the difference between the product and the first term when the series is decreasing. To find the first term-Multiply the number of the differences by the difference, and subtract the product from the last term when the series is increasing, or add the product to the last term when the series is decreasing. EXERCISES. Ex. 1. The first term of a series is 1, the common difference is 2, and the number of the terms is 12, what is the last term? Answer 23. Ex. 2. The first term of a series is 1, the 12 th is 23, and the number of the terms is 12, what is the sum of the series? Answer 144. Ex. 3. If the extremes be 3 and 19, and the number of the terms 9, what is the common difference, and the sum of the whole ? Answer Diff. 2, Sum 99. Ex. 4. If a person travelling for 10 days, makes his first day's journey 20 miles, and increases his rate 5 miles each day, what distance will he have travelled by the end of his journey? Answer 425 Miles. Ex. 5. The hour circle of astronomical clocks is divided into 24 hours, and the hours are usually reckoned to 24 from noon of one day to noon of the following day. If, therefore, a clock of this sort were furnished with a striking apparatus like a common clock, how many strokes would it strike in the course of one circuit? Answer 300. Ex. 6. What will be the value of 100 books, reckoning the first at 3 farthings, and each succeeding book at an increased value of 3 farthings? Answer £15 15 7. Ex. 7. If 100 stones be placed in a right line, exactly a yard asunder, and the first a yard from a basket, what length of ground must a person go who gathers them up singly, and returns with them one by one to the basket? Answer Miles 5 1300 yds. Ex. 8. What is the last term, and also the sum of a decreasing series, in which the first term is 5, the common difference, and the number of the terms 8? Answer Last term 43, Sum 383. Ex. 9. What is the last term of a decreasing series, in which the first term is 8.368, the number of the terms 6, and the common difference 1.014? Answer 3.298. Ex. 10. What is the sum of an increasing series, in which the first term is 8.368, the number of the terms 6, and the common difference 1.014. Answer 65.418. GEOMETRICAL PROGRESSION. Three or more numbers are said to be in Geometrical Progression, when each term in the series is produced from the preceding by the operation of the same ratio or multiplier; this multiplier being a whole or mixed number, that is greater than unity, when the series is increasing, and a fractional or decimal number, that is less than unity, when the series is decreasing. In the numbers of any continued Geometrical Progression the product of the extremes is equal to the product of the means, when the number of the terms is even; and to the square of the mean term, when the number of the terms is odd: hence the geometrical mean between two terms is equal to the square root of their product; as with the numbers 4 and 64, 16 is the mean proportional, being the square root of the product 256; and this mean proportional, may be considered as being both the second and the third terms of a simple proportion, for 4 is to 16 as 16 is to 64. When a number is divided into two such parts, that the ratio of the whole to one of the parts, is the same as the relation of that part to the remaining part, the number is said to be divided into extreme and mean ratio; of which parts, the larger is found by multiplying half the given number by the difference between the square root of 5 and unity, or by the square root of 5 less 1: thus, as the square root of 5 is nearly 2.236068, the half of any number multiplied by the factor 1.236068, gives nearly the greater of the two parts of the number which is to be divided into extreme and mean ratio. thus for 20 10 × 1.23606812.36068 is the greater part nearly 20-12.36068: 7.63932 is the less part for 20 12.36068 :: 12.36068 : 7.63932 If the first term of a series be unity, the succeeding terms will be the involved powers of the ratio : thus, in 2 series ascending and descending, if the ratios are 2 and, the terms are If the series commence with any other number than unity, the value of any term will correspond with the product of the first term multiplied by such a power of the ratio, as corresponds with the number of the terms less one: thus, if the first term be 5, and the ratio 2, then the 7 th term will be equal to 5 multiplied by the 6 th power of 2, or by 64; that is, to 320; for the series will be 1 st 2 nd 3 d 4 th 5 th 6 th 7 th This corresponding power of the ratio is called its index, and in their application to the terms of a series, the indices commence with 0 and increase by 1; hence with the preceding terms, we have, as The indices of powers formed in this manner, are called the Logarithms of the powers considered as natural numbers; they possess the property that by the addition or subtraction of the indices or logarithms of two natural numbers, the amount forms the logarithm or index of the product or quotient of the natural numbers. Thus, with the natural numbers 4 and 16 in the preceding formula, the indices or logarithms are 2 and 4; the sum of these is 6, which is the logarithm of their product 64. On the contrary, taking 6, which is here the logarithm of 64, and subtracting 4, the logarithm of 16, the remainder 2 is the logarithm of 4, the quotient of the division of 64 by 16. The involution and evolution of numbers are performed with their logarithms, by multiplying or dividing by the number of the power or root. Thus, to cube 4, if we take its logarithm 2 and multiply it by 3 we have 6, which is the logarithm of 64, or the cube of 4. So in the reverse, to extract the cube root of 64, if we take its logarithm 6 and divide it by 3 we have 2, which is the logarithm of 4, the cube root of 64. The series of the powers usually employed for the tables of logarithms, is that which corresponds with the natural combinations of numbers into tens, the logarithms and numbers being, |