gent, &c. (as in the Scheme) we shall have the Sine, Tangent, &c. to every ten Degrees in the Quadrant: and the fame way we may have the Sine, Tangent, &c. to every fingle Degree in the Quadrant, Quadrant, by dividing it into 90 equal Parts beginning from B, and drawing the Sine, Tangent, &c. to all the Arches beginning at the fame Point B. By this Method they draw the Lines of Sines, Tangents, &c. of a certain Circle on the Scale; for after drawing them on the Circle they take the Length of them, and fet them off in the Lines drawn for that purpose. The fame way, by fuppofing the Radius of any Number of equal Parts, (fuppofe 1000, or 10,000, &c.) 'tis plain the Sine, Tangent, &c. of every Arc muft confift of fome Number of thefe equal Parts, and by computing them in parts of the Radius, we have Tables of Sines, Tangents, &c. to every Arch in the Quadrant, called Natural Sines, Tangents, &c. and the Logarithms of these gives us Tables of Logarithmic Sines, Tangents, &c. To understand the Nature of which, and the Method of using them, you must know that Logarithms are only artificial Numbers, contriv'd to avoid long Operations in natural Numbers, each of which has a Logarithm belonging to it. Their Nature is fuch, that Addition of them answers to Multiplication in natural Numbers, and Subtraction anfwers to Divifion; that is, when two Numbers are propos'd to be multiply'd into one another, if we take the Logarithms anfwering to the Numbers and add them together, the Sum will be the Logarithm anfwering to the natural Number, which is the Product of the two Numbers propofed. Again, when one Number is proposed to be divided by another, if from the Logarithm of the Dividend we fubtract the Logarithm of the Divifor, the Remainder fhall be the Logarithm of the Quo tient. Now to apply this to practice: The firft Table at the end of this Book, contains the Logarithms of all the Numbers from 1 to 10000; the Columns mark'd at the top with (N) contain the natural Numbers, and the adjacent Columns contain the Logarithms of these Numbers. So to find the Logarithm of any Integer Number between 1 and 10,000, we must look in the Columns mark'd with N at the top, till we find the Number propos'd; and that standing on the fame Line with it in the adjacent Column is the Logarithm required. Example. Let it be required to find the Logarithm of 365; by looking in the Table according to the above Direction, I find it to be 2.56229. The Reverse of this, viz. Given a Logarithm, to find from your Tables the natural Number anfwering thereto, is perform'd by looking into the Columns mark'd with Logarithm at top, for that which is either equal or nearest to the one propos'd, and the Number anfwering to it in the adjacent Column is that required. Example. Let it be required to find the natural Number answering to the Logarithm 2.56229, by proceeding according to the above Direction I find it to be 365. Again, if it were required to find the Logarithm of a Number, having fome Decimals in it. In order to do this, you may obferve in the Table of Logarithms, that the Logarithm of 10 is 1, that of 100, 2; and of 1000, 3, &c. and the Logarithms of all the intermediate Numbers between 10 and 100, have 1 for the integral Part of each, and all those between 100 and 1000 have 2 for their integral Part, and fo on, which are called their Indices. Now because any Number confifting of both integers and decimals, is equal to the Quotient of the whole confider'd as an Integer divided by the Denominator of the decimal Part; and fince by the Nature of Logarithms, Subduction in them anfwers to Divifion in other Numbers; therefore it follows, that when a Number is given confifting both of in tegers tegers and decimals, we can find the Logarithm answering thereto in the following manner: viz. Find the Logarithm of the whole confider'd as an Integer; then from that take the Logarithm of the Denominator of the decimal Part, or (which is the fame) from the Index of the Logarithm of the whole confider'd as an Integer, fubtract a Number lefs by Unity than the Number of Places in the Denominator of the fraction, and the Remainder will be the Logarithm required. Example 1. Suppose you were to find the Logarithm of 36.5; to do this you must first look for the Logarithm of 365, which is 2.56229, then because 10 is the Denominator of the decimal Part of the propos'd Number, and 1.0000 its Logarithm, therefore from 2.56229 take 1.0000, and there remains 1.56229 the Logarithm required. Example 2. And to find the Logarithm of 6.543. First find the Logarithm of 6543 confider'd as an Integer, which by the Tables you will find to be 3.81578; then fince 3.0000 is the Logarithm of 1000 the Denominator of the fractional Part, therefore from 3.81578 take 3.0000, and there will remain 0.81578, which is the Logarithm required. The Reverse of this, viz. the Logarithm of a Number confifting of integers and decimals being given to find that Number, is perform'd according to the following Method. Rule. Look in your Table of Logarithms (without regarding the Indices) for that whofe decimal Part is equal or nearly equal to the decimal Part of the Logarithm propofed; then fubtract the Index of the former from that of the latter; and lastly divide the Number anfwering the Logarithm found in your Tables, by a Number confifting of an Unit, and as many Cyphers as there are Units in the difference between the two Indices; or, which is the fame, cut off as many Figures (beginning at the lowest place) of the Number answering to the Logarithm in your Table, as there are Units in the difference of the Indices, and the Number last found will be that required. Example. Suppofe it were required to find the Number answering to the Logarithm 2.73608. In order to do this, I look in the Table of Logarithms (without minding the Indices) for that whofe decimal part is equal, or nearly equal, to 73608, the decimal part of the Logarithm propos'd, and I find it to be 3.73608; from the Index of which, viz. 3, I take 2, the Index of the propos'd Logarithm, and there remains ; laftly, I divide 5446, the Number anfwering the Logarithm found in the Tables, by 10, and the Quotient 544.6 is the Number required. The Reason of this and the preceeding Rule, is plain from the very Nature of Logarithms. From what has been faid on this Head we may eafily folve the following Problems by the Logarithms: viz. Prob. 1. Given two Numbers, as 25.6 and 134, to find the product of their Multiplication. To folve this by the Logarithms, I first look for the Logarithm of 25.6 which I find to be 1.40824, then for that of 134 which is 2.12710; then I add these two Logarithms together, and their Sum is 3.53534, which is the Logarithm of their product; fo I look in my Table for the Number anfwering to 3.53534, and I find it to be 3430, which is nearly equal to the product of 25.6 into 134. Again, if it were required to find the product of 36 into 234, I proceed as in the last Example, and the Operation is as follows: 2.36922 the Logarithm of 234 Sum 3.92552 the Logarithm of their Product. E 2 which, |