Trigonometria Britannica

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Chapter Sixteen

Concerning the Logarithms of Sines, Tangents, and Secants.

Logarithms are the obliging equidistant numbers of proportional numbers: with the help of which, being given any three numbers whatsoever, the fourth proportional being found more easily by Addition and Subtraction alone. But the whole of Trigonometry has been founded in the finding of the fourth proportional number in any of these Canons of Sines, Tangents, or Secants. Therefore with these Tables we attach the Logarithms, as with these the [4th proportional] number sought being produced with the least trouble.

But especially the Logarithms of the Sines have been sought conveniently; then for the Tangents and Secants.

The different kinds of Logarithms for the same numbers will be able to be applied: as being shown by me in the first chapter of the Arithmetica Logarithmica. The most distinguished man, the Baron of Merchiston who first found these numbers, had explained these [i.e. Napierean logarithms] which he had prepared at that time. And that more of the same have followed, stepping in the same footsteps; from which number Benjamin Ursinus has produced a great and praiseworthy work, these being applied to Degrees, minutes, and sixths of minutes. I truly being helped with encouragement by this first inventor himself, I considered the applications of other Logarithms, which have many a use, much easier and more excellent [i.e. those to base 10].

The Logarithm of the Radius of the circle or the total Sine, by me being set 10,00000,00000,0000. The Radius for this Logarithm being congruent being set 100000,00000, of which the Characteristic by Chapter 4 of the Arithmetica Logarithmica is 10. The Characteristic of the remaining Sines truly is 9; until we arrive at the sine of 5:44', or 5.73 , from which place as far as to 0:34, or 057 the Characteristic is 8. Then truly 7 as far as 0:3 or 0.05. And finally the Characteristic is 6 at the beginning of the Table. Because with the decreasing Sines, which have fewer places with any sine you wish, the least Characteristic is to that [one]. But the number of places in this Table was above the Characteristic, as we would have the Sines themselves more accurate. Finally truly five places have been added on to the Sines, so that the minutes would not seem to be absent from the perfection of these.

For any number you wish, the way to consider its own Logarithm being propounded in Chapter 14 of the Arithmetica Logarithmica: following which (by dividing the Quadrant into 72 equal parts) for the Sines of these particular parts their own Logarithms have been furnished; and the other numbers being augmented by quinquisection, as they shall be 360, then 1800, finally 9000. With the individual Degrees being assigned 100 Logarithms. The method of quinquisection with Logarithms is the same which has been with Tangents and Secants, with which the mean Differences are being corrected by subtraction only.

If we should wish to display Logarithms to thousandths of Degrees, the Sines have to be prepared, which (by the division of the Quadrant into 144 small equal parts) of these separately are convenient: and the Logarithms themselves ought have been fitted out for these, and of these the number being increased by Quinquisection as before, so they arrive at 720, 3600, 18000, 90000.

But as there shall be a very large inequality in the Differences in the Logarithms of the Sines at the beginning of the Quadrant, it will scarcely be able with this Quinquisection method to show the accurate Logarithms of the first degrees. It is therefore this defect, and for no other reason this following Proposition ought to compensate for the Sines themselves, by the golden rule of proportionality; so for the Logarithms of the Sines, it will serve most conveniently for finding [these] by Addition and Subtraction alone.

2. If the Sines shall have been given, either from the beginning of the Quadrant to 45 Degrees: or from the same Degree as far as the end: the remainder of the Sines being able to be found by the golden rule. For they are the Sine of any Arc whatsoever and half the radius of the circle, the mean proportionals between the Sine of the arc being halved, and the Sine of the complement of the same half. As will be shown in this Diagram

Let DE be the Sine of 56:0' Degrees, BC the Sine of 28:0', AC the Sine of the Complement 62:0', AO half of the Radius. BD being drawn the Chord of 56:0', and CF perpendicular to the Radius. ABC, ACF will be similar triangles, and the lines AB, BC : AC, CF proportionals. In like manner AB, AO: DE, CF. And therefore the rectangles BC, AC; AO, DE; AB, CF are equal; and the sides of the equal rectangles are in reciprocal proportion: BC to AO, as DE to AC.1

If the Sine shall be given from the first Quadrant as far as 45 Degrees, the remainder of all the Sines will be able to be found by the proportional rule, as you see here, if we except the Sine of 60 Degrees.

However if the Sines will have been given from 45 Degrees to the end of the Quadrant, the remainder from the beginning as far as 45 Degrees being found by the same method.

3. And besides this method for the Sine itself; so the Logarithms of these but with much less trouble being found. For with the Sines of half the Quadrant being found, and (by Chapter 14 of Arithmetica Logarithmica) for the Logarithms of these: of the remaining Sines all the Logarithms for the Quadrant, by Addition and Subtraction, or by Addition alone being found. For with universal proportionality, the [sum of the] Logarithms of the extremes being equal to the [sum of the ] Logarithms of the means, and from four Sines in proportion, three together with the Logarithms of these will have been given, the fourth and of this the Logarithm most surely and easily becomes known. Three examples for the method of working, and the truth should have been established with more Propositions, I have been of the opinion, ought to be added. In the first the Logarithm of the first being taken from the sum of the means. With the two remaining in the first position being placed the Arithmetical Complement, and the Logarithm of the number sought arriving at the fourth place. See Ch. 15 of Arith. Logar. we advise.
The Logarithms of the Sines following for the individual parts of the Quadrant being divided into 72 equal parts, which by Quinquisection (where the work will have been aided by the preceding preposition) will become 360, 1800, 9000

Note On Chapter Sixteen

1 Thus: AB/AO = DE/CF; and
AB/BC = AC/CF. From which AO.DE = BC.AC, and BC/AO = DE/AC. Hence,

sin(theta/2)/(R/2) = sin(theta)/(R cos(theta/2),

a variant of the double angle formula for the sine; or, as Briggs would have it:

sin(theta/2)/sin(30) = sin(theta)/sin(p/2 - theta/2)].

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Ian Bruce January 2003