M. Q. R. Answer, 69 1 75.11. One word of advice to the young surveyor, who is coming forward to be useful in his occupation, will close the appendix. In the choice of assistants to perform practical operations, never call to your aid Sir Richard Rum. He frequently changes his name to brandy, gin, whiskey, &c. cherous, and he causes the head to whirl, the and the foot to stumble. He is treabody to reel, By his might, the strong man has fallen, and the promising youth has been brought to an untimely grave. If you employ Sir Richard, your columns of latitude may differ too much for correct work, and your columns of departure may be still worse. He never can render you the least service in establishing your reputation as an accurate surveyor, but there is danger that he may do much to destroy it. NOTE. Such theory as is more curious than useful, however correct it may be, has been excluded from the appendix. The plainest methods have been selected. On some points, the Author has been more minute, on account of the fact that many copies of this book are bought by men who do not expect to be surveyors, and who do not place themselves under instructors. If some repetition is to be found in this work, the learner will find less fault with it than the critic. Hebron, (Conn.) June, 1835. LOGARITHMS. THE learner, who, for the first time, becomes acquainted with the wonderful properties of Logarithms, may be not a little surprised to find himself introduced to a system of numbers, so new in their nature, and which, surpassing all his former knowledge of figures, afford so many facilities for shortening the labour and lessening the difficulty of arithmetical calculations. He will admire to find, that by help of these, the labour of hours, and in some calculations, even the labour of days, may be reduced to as many minutes! The invention of Logarithms was justly regarded as a favour from heaven;" because, in many departments of science, essential to the happiness of man, they have saved him ages of toil. Although it does not come appropriately into the design of a work like this, to enter minutely into the history of their invention, nor the yet more difficult process by which they were originally constructed, yet a familiar explanation of their properties and uses, adapted to the apprehension and wants of the practical surveyor, is necessary, in order to his making a proper application of their great advantages in practice. Logarithms, then, we may first observe, never stand for the numbers themselves, of which they are composed, but invariably for other numbers, of which they are only the representative exponents, or indices. Their great utility in arithmetical operations, consists, chiefly, in this, -that addition takes the place of multiplication, and substraction that of division. That is, to multiply numbers, we have only to add their logarithms; to divide, we have only to substract the logarithm of the divisor from that of the dividend; to raise a number to any power, we multiply its logarithm by the exponent of that power; and to extract the root of any number, we merely divide its logarithm by the number expressing the root to be found. The constant number upon which the tables in common use are constructed, and which is called the base of the tables, is 10; and every conceivable number, large or small, integral, mixed, or decimal, is considered as some ascertained power or root of 10. Indices. 101 the first power of ten is Numbers. 10, whose exponert is 100, whose exponent is 1,000, whose exponent is 10,000, whose exponent is 100,000, whose exponent is 106 the sixth power of ten is 1,000,000, whose exponent is Logarithms. 1. 2. 3. 4. 5. 6. NOTE. It may be remarked, that the first power of any number, is that number once repeated, or it is the number itself: The second power of any number is the product of that number multiplied once by itself: The third power of a number, is the product of the number multiplied twice by itself; the fourth power of a number, is the product multiplied three times by itself, &c. The index denoting the power, is called, in common arithmetic, the exponent of that power; and is, in other words, the logarithm of the power. Logarithms, then, are the Exponents of a series of powers and roots. In the above series, the logarithms indicate how many ciphers belong to their corresponding numbers. Thus, the logarithm 1 stands for 10, phers; the logarithm 3 for 1000, or 1 and three ciphers, &c. Now, if we multiply 10,000 by 100, the product will be 1,000,000, whose logarithm is 6; but to obtain this, we need only add the logarithms 2 and 4, which stand opposite the numbers to be multiplied. On the contrary, if we divide 1,000,000 by 100, the quotient will be 10,000, whose logarithm is 4 but to obtain this, we need only to substract 2, the logarithm of the divisor, from 6, the logarithm of the dividend. Again, the square of 1000, that is, the product of 1000 multiplied by itself, is 1,000,000, whose logarithm is 6; but to obtain the square of 1000, we need only double its logarithm 3. On the other hand, the cube root of 1,000,000 is 100, whose logarithm is 2; but this is obtained by dividing 6, the logarithm of the given number, by 3, the index of the root. Hence it is manifest, that the protracted labour of multiplying or dividing one large number by another, the tedious evolution of roots, and the various mistakes incident to long operations, may be almost entirely obviated by the use of logarithms. As the logarithm 1 is always 0, and that of 10 is but 1, the logarithms of all numbers below 10, will be decimals; and as the logarithms in the common system increase regularly by 1, according to the integral powers of 10, it follows that the logarithms of all numbers between 10 and 100, will be more than 1, but less than 2-that is, they will be 1 and a decimal; the logarithms of all numbers between 100 and 1000, will be between 2 and 3-that is, they will be 2 and a decimal; and the logarithm of all numbers between 1000 and 10,000, will be between 3 and 4--that is, 3 and a decimal. A logarithm generally consists of two parts; a whole number, and a decimal. This whole number, or integer, is called the characteristic, or index, of the logarithm, and is always one less than the number of integral figures in the natural number, whose logarithm is sought. As the index of the logarithm is omitted in the tables, it is important to recol lect the principle, or rule, by which it is to be supplied, whenever it is wanted in calculation. Thus, the logarithm of 8 is 0.903090. Here, the number (8) consists of but one figure, and the index of its logarithm, being one less, must be 0. Again, the logarithm of 16 is 1.204120. Here, the given number (16) consists of two figures, and the index of its logarithm, being one less, must be 1. Again, the logarithm of 640 is 2.806180. Here, the given number (640) consists of 3 figures, and the index of its logarithm, being one less, must be 2, &c. The rule holds universally true, that the index of a logarithm is always one less than the number of integral figures in the natural number, whose logarithm is sought. The same rule holds in mixed numbers. The logarithm of 6.40 is 0.806180, the same as for 640, (see the last example,) differing only in the index. Here, the integral part (6) of the given number, consists of but one figure, and the index of its logarithm, being one less, must be 0. And, generally, having obtained the logarithm of any number, large or small, we have only to change the index, agreeably to the above rule, in order to obtain the logarithm of every other number, consisting of the same significant figures, whether they be integral, fractional, or mixed. Thus: When the natural number is less than 1, the index of its logarithm becomes less than 0, or negative; and is indicated by placing the sign, just before, or above it. Suppose it were required to affix the proper index to the logarithm of .000007596. Here, the number of cyphers on the left, including the decimal point, is 6, which being fitted with the negative sign, -, becomes the proper index of the logarithm. And universally, The negative index is always equal to the number of ciphers on the left, including the decimal point. Before any one can avail himself of the great advantages of logarithms, in expediting the operations of Arithmetic and Trigonometry, he must become so familiar with the tables, that he can readily find the logarithm of any number; and, on the other hand, the number to which any logarithm belongs. DIRECTIONS FOR TAKING LOGARITHMS AND THEIR NUMBERS FROM THE TABLE. NOTE. In the common tables, the Indices to the logarithms of the first 100 numbers are inserted. But for all other numbers, the decimal part only of the logarithms is given while the index is left to be supplied, according to the principles already laid down. PROBLEM I.—To find the logarithm of a number between 1 and 100. RULE. Look for the proposed number on the left; and against it, in the next column, will be the logarithm with its index. EXAMPLE. The logarithm of 50 is 1.698970. The logarithm of 89 is 1.949390. PROBLEM II. To find the logarithm of any number between 1 and 1000 or of any number consisting of not more than three significant figures, with ciphers annexed. RULE. Find the given number in the left hand column of the table, and directly opposite, in the next column, is the decimal part of its logarithm, to which apply the index, as already taught. EXAMPLE. The logarithm of 140 is 2.146128. The logarithm of 781 is 2.892651; of 358 is 2.553883; of 974 is 2.988559. The decimal part only of these logarithms are found in the table; the index 2, was affixed to each, because the given numbers consisted, each, of three integral figures. If there had been ciphers annexed to the significant figures of the given numbers, as 1400, 35800, &c., their logarithms would have been precisely the same, with the exception of the index only; and consequently, would be found in the same place in the table. Thus: The log. of 1400 is 3.146128. of 35800 4.553883. The log. of 781000 is 5.892651. of 9740000 6.988559. Here the decimal part of the logarithm is the same as before; while the index has been increased as many units, as there are ciphers annexed to the given numbers. This rule will hold good in all similar cases. PROBLEM III. To find the logarithm of any number consisting of four figures, either with or without ciphers annexed. RULE. Look for the three first figures, on the left hand, and for the found opposite the three first figures, and in the column which, at the head, is marked with the fourth figure. By reference to the table, it will be seen, that each page contains ten columns of logarithms, which are severally numbered from 0 to 9. The first column, alone, contains six figures; while every other column has only four figures: but it is to be always remembered that the two first figures of the left hand column, are common to each of the other columns, and were omitted only to avoid repetition. These two initial figures, therefore, are to be prefixed to each of the other four, since every logarithm, in our table, consists of six figures, besides the index. EXAMPLE. The log. of 3657 is 3.563125. The log. of 6704 is 3 826334. of 8512 is 3.930032. of 5696 is 3.755570. In the last example, as it will frequently happen, the two initial figures (93) of the logarithm, are not found, in the same line, with the given number, (851,) but in the next below it:-And, universally, whenever the third figure of the logarithm changes, from 9 to 10, the cipher only is retained in the column, while the one is carried down to the next lower initial, on the left. To guard against a mistake here, points have been substituted in place of ciphers; and wherever these points are found, the ciphers are to be reinstated, and the two initials iaken from the line below. To be more particular, in the above example; on turning to page 14 of the logarithms, and against 851, the learner will find 92 for the two initial figures, which he must prefix to the other four figures in the first and second columns, but no farther. There he must stop, and taking the two initial figures in the line below, against 852, carry them up to the third column, where the dots commence, and prefix the same to each of the remaining columns: and so in all similar cases. PROBLEM IV. To find the logarithm of a number consisting of five or six figures. RULE. Find the logarithm of the first four figures of the given number, as taught in the last problem. Take the remaining figures and multiply them into the number standing opposite, in the outside column, headed D; from the right of the product, reject as many figures as you multiplied by, and add what is left to the logarithm previously found. This sum, being fitted with a proper index, will be the logarithm required. EXAMPLE. Required the logarithm of 45263. Thus 4.655715 The difference D is 96, which being multiplied by 3 gives 28.8 Logarithm of 45263 required, 4.655743 EXAMPLE 2. Required the logarithm of 758936. Thus The logarithm of 758900 is 5.880185 The difference D is 57, which, being multiplied by 36, gives 20.52 Logarithm of 758936 required, 5.880205 NOTE. This process of finding the logarithms of large numbers, supposes that they increase in the same ratio as their numbers, which is not strictly true, though sufficiently near the truth for general practice. It may be remarked, however, that these ratios approach that of equality, the larger the numbers, and the less they differ from each other. The column marked D, contains the average mean differences of the ten logarithms against which they stand, and, consequently, do not al |