Chapter Fifteen

With the Canon of Sines for Hundredths or Thousandths of Degrees, the Canons of Tangents, Secants, and of Logarithms are being provided with the same parts.

Prop.1. Tangents and Secants are most conveniently being found by the Rule of Proportion. For any Sine is to the Sine of the Complement : as The Radius to the Tangent of the same Complement.

From this Proposition alone any of the whole Quadrant of Tangents will be able to be found.

Prop. 2. The Radius is the mean proportional between the Sine of the Arc and the Secant of the Complement of these as you please.

From this Proposition any Secants you wish will be able to be found.

Prop. 3. The Radius is to the Sine of any Arc you please: as the Secant of the same to the Tangent.

Prop.4. The Radius is the mean proportional between the Tangents and of the Complement of these Arcs as you please.

If by dividing the Quadrant into 144 parts, by the first proposition of these, the tangents of half the Quadrant or with the first 72 equal parts being found appropriately; the Tangents of the remaining parts will be able to be found, and the Secants of all the others, by addition alone. As by the following Propositions being demonstrated.

Prop. 5 The Secant of any Arc you please, being equal to the [sum of the] tangents of the same Arc and half of the Complement1.

Let the Angle EAD be 23:0', and with the line GEF touching the periphery in the point D. GE, EF being taken equal to the line EA. GAF will be right, and DAF, EGA, EAG equal among themselves, and EAG half the Compliment [of EAD] EAB. But EF is equal (equal from the construction of the line AE to the secant of the angle EAD) to [the sum of] the Tangents ED of the angle EAD given, and DF of the angle FAD of half the Complement EAD 23:0', EAB 67:0' the Complement , DAF 33:30'.

Prop. 6 The Secant of any Arc you wish, being added to the Tangent of the same, being equal to the Tangent of the Arc being composed from the given arc and half of the Complement. [As AE + ED = DG: Figure 15-3].

For let the Angle EAD be 23:0', the Complement 67:0'; half of the Complement 33:30'. The Arc being composed 56:30'

Prop. 7. The Tangent of any Arc you wish by being taken from the Secant of the same, there is left the Tangent of half the Complement. Prop. 7.

The Tangent being doubled of any Arc you wish, by being added to the Tangent of half the Complement, will be equal to the Tangent of the Arc composed from the given arc and from half of the Complement. Prop. 8. [For 2ED + DF = EF + ED = GE + ED = GD].
For by Prop. 5 the Secant is to be equal to the [sum of] Tangents of the same arc and of half the Complement. And therefore by Prop. 6 if to the double of the tangent of the given arc being added to the Tangent of half the Complement, the sum will be equal to the Tangent of the arc composed. Therefore with the given Tangents, with the individual parts with convenience from the first half quadrant ; the Tangents will be able to be found for the remaining parts for all the Quadrants, by addition alone following this Eighth Proposition.

And by this method the Tangents of the 144 separate parts of the Quadrant being found; The Secants of the other parts will be found following Prop. 6 by addition alone. Indeed if the Tangent of half the Complement of that arc (of which the Secant is being sought) will have been given. As

We have therefore a handy enough way for finding the Tangents and Secants from the 72 parts of the Quadrant. It will be possible to increase the number of these by quinquisection (as of the Sines before): at first to 360, then to 1800, by the third to 9000, thus as with individual degrees the whole of the Quadrant may be considered 100 Tangents, and the same number of Secants. The method of working is nearly the same which was expounded before on page 38 [i.e. the interpolation scheme of Chapter 12.]

From this so much is placed between; because there with the correction of the Differences for the Sines as equally we used Addition as Subtraction, but with these it is fitting to use Subtraction alone.

If we should wish to construct the Canon to thousandths of Degrees; the secants for the remaining parts of the Quadrant, which by the recent method were unable to be placed, should be sought by the second Proposition of this chapter, for as with Secants so with Tangents for the 144 parts of the Quadrant being computed quickly. Then the number of these being increased as before to 720 parts, secondly to 3600 parts, and thirdly to 1800, and finally to 90000.

¹ Here is the diagram: