Arithmetical Spyglass and Teacher's Assistant; Intended as a Key and Supplement to the Different Works on Arithmetic, for the Use of Schools and Academies (Paperback)

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1842 edition. Excerpt: ...those quantities proportional. 3. If four quantities are proportional, the rectangle of the means, divided by either extreme, will give the other extreme. 4. The products of the corresponding terms in continual proportions, are also proportional, 5. If three numbers be in continued proportion, the square of the first will be to that of the second, as the first number to the third. 6. In any continual proportion, the product of the two extremes, and that of every other two terms, equally distant from them, are equal. 7. The sum of any number of quantities, in continued proportion, is equal to the difference of the rectangle of the second and last terms, and the square of the first, divided by the difference of the first and second terms. fl As the last term, or any term near the last, is very tedious to be found by continual multiplication, it will be very necessary, in order to ascertain it, to have a series of numbers in arithmetical proportion, called indices, or exponents, beginning either with a cipher, or a unit, whose common difference is one. When the frst term of the series and the ratio are equal, the indices must begin with a unit; and, in this case, the product of any two terms is equal to that term signified by the sum of their indices. Thus, 1,2, 3, 4, 5, 6, &c., indices, or arithmetical series. And 6-)-6=12, index. Then 2, 4, 8, 16, 32, 64, &c., geometrical series (leading terms) of the twelfth term. And 64X64=4096, the twelfth term. But, when thefirst term of the series and the ratio are different, the indices must begin with a cipher, and the sum of the indices, made choice of, must be one less than the number of terms, given in the question; because 1 in the indices stands over the second term, and 2, in the...