This chapter contains sections 163 to 189
Section 163 edit
The definition of the tangent and the secant: If a line that is half the diameter in length is extended from the center to one end of an arc, and at this point another line is drawn outside of the circle vertically at a right angle to this line, the vertical line is called the 'tangent'. A line drawn from the center of the circle to the other end of the arc, extending in such a way that it eventually meets the vertical line is called the secant. In this example:
Line BC is half the diameter, line AC is the vertical tangential to the arc GC, and line AB is the secant of arc GC. There already exists a chart of all tangents and their secants for every degree in all arcs and also angles - since tangents and secants of arcs are also tangents and secants of angles, and follow the same patterns as written for sines.
Section 164 edit
A right angle or quarter arc has no tangent or secant since the secant drawn outward will never meet the vertical. For example, in arc FC above, the tangent CA and the secant BF never meet since they are parallel lines, as was written in section 19.
Section 165 edit
The tangent and secant are 'partners' with an arc and its remainder. For example, tangent AC and secant AB are 'partners' with arc GC and arc GH. They also have the same relationship with the angle and its remainder.
Section 166 edit
The complementary tangent and secant are the tangent and secant that complete the arc to 90 degrees. If the arc is greater than 90 degrees, the degrees of complementariness are the tangent and secant of an arc with a measure of degrees over and above 90. For example, the tangent FI is the tangent for arc FG, which is complementary to the tangent of CA, which is the tangent for arc CG and arc HG. Similarly, secant BI is complementary to secant BA for the same reason.
Section 167 edit
The ratio of the tangent to the maximum sine is proportional to the ratio of the maximum sine to the complementary tangent. The proof to this can be seen in the previous figure, where the ratio of [vertical tangent] line AC to [length of maximum sine, half the diameter] line BC is proportional to the ratio of [maximum sine] line BF to [complementary horizontal tangent] line FI. [Refer to section 92]
Section 168 edit
The ratio of the tangent to the secant is proportional to the ratio of the maximum sine to the complementary secant. For example, the ratio of line AC to line AB is proportional to the ratio of line BF to line BY. Similarly, the ratio of the maximum sine to the secant line is proportional to the ratio of the complementary tangent line to the complementary secant line. For example, the ratio of line BC to line AC is proportional to the ratio of line AF to line BI.
Section 169 edit
The ratio of the complementary sine to the sine is proportional to the ratio of the maximum sine to the tangent. The proof of this is as follows: line BD is equal to line ED, which is the complementary sine of arc GC. Line GD is the sine of arc DB, and lines GD and AB are parallel since they are both right angles from line BC. Since this is so, the ratio of line BD to line GD is proportional to the ratio of line BC to line AC. Furthermore, the ratio of the sine to the complementary sine is proportional to the ratio of the maximum sine to the complementary tangent line, for if one compares BG to arc FG one will find that line BD is the sine of the arc, and line GD is the complementary sine, with line AG being the complementary tangent line.
Section 170 edit
The ratio of the cosine to the maximum possible sine line is equivalent to the ratio of the maximum sine line to the secant that divides the arc The proof of this is similar to what was written in a previous section [92] where the proportion of all these are explained in relation to arc FG. [Referring to the figure in section 163, the arc FG creates sine EG and cosine GD. These together create two triangles facing each other, BFI and BGD. These two triangles were shown to be in proportion to each other, so that vertical line GD is to vertical line BF as diagonal BG is to diagonal BI.]
Section 171 edit
The proportion of the sine to the maximum sine line is equivalent to the proportion of the tangent to the secant. The proof of this is that the proportion of line GD to BG is equivalent to the proportion of line AC to AB.
Section 172 edit
The square of the secant is equal to the sum of the squares of the maximum sine line and the tangent line. In the above figure the square of AC is equal to the sum of the squares of lines BC and AB, as was written in section 75.
Section 173 edit
In right triangle [ABC], the proportion of the tangent [AC] of angle A to the maximum sine [BC] is equivalent to the proportion of the line opposite angle A [line GD] to the line perpendicular to it [line BD]. Similarly, the proportion of the maximum sine [BC] to the secant of this angle [BA] is equivalent to the proportion of the line adjoining the angle [BD] to the hypotenuse [BG]. The proof of this can be found in triangle ABG, where line BC is considered the maximum sine, and an arc drawn with this diameter [with its center at point A] would form arc GC, with angle ABC having the tangent of AC and the secant AB. Therefore, if line AC [per commentary] is considered the maximum sine then side BC would become the tangent of angle BAC and line AB would be its secant.
Section 174 edit
[Returning to the methods of deducing a triangle's quantities...] Now see before you that there is another method whereby you may know all remaining quantities [aside from those previously discussed starting in section 110], if the figure is such that two quantities aside from the right angle are known in a right triangle. This approach will make use of tangents and secants. The first method: If the lengths of the vertical and base are known, meaning lines AB and BC are known, the proportion of side AB to side BC will be found to equivalent to the proportion of the maximum sine to the tangent of angle BAC. Once the angle BAC is known, realize that a) the proportion of the maximum sine to the secant of angle BAC is equivalent to the proportion of side AB to AG, b) that the proportion of the tangent of angle BAC to the secant of BAC is equivalent to the proportion of side BC to AC, and c) that the proportion of side BC to AB is equivalent to the proportion of the maximum sine to the tangent of angle ACB. Following this one may calculate that a) the proportion of the maximum sine to the secant of angle ACB is equivalent to the proportion BC to AC, and b) that the proportion of the tangent of angle ACB to the secant of ACB is equivalent to the proportion of side AB to side AC.
Section 175 edit
The second method: If the length of the vertical line and the hypotenuse are known, for example, if lines AB and AC are known, then the proportion of line AB to line AC is equivalent to the proportion of the maximum sine to the secant of angle BAC. Following this you will find that the proportion of the maximum sine to the tangent of angle BAC is equivalent to the proportion of line AB to line BC, and the proportion of the secant of angle BAC to the tangent of angle BA is equivalent to the proportion of side AC to side BC.
Section 176 edit
The third method: If the base line and the hypotenuse are known, for example lines AB and BC, then base can be set up in proportions in the same manner as was done with the vertical side in the previous section.
Section 177 edit
The fourth method: If and BAC and side AB are known, recognize that the proportion of the maximum sine to the tangent of angle BAC is equivalent to the proportion of side AB to side BC, or that the proportion of the tangent of angle ACB (which is the cosine of angle BAC) to the maximum sine is equivalent to the proportion of side AB to side BC. Additionally, the proportion of the maximum sine to secant of angle BAC is equivalent to the proportion of side AB to side AC, and the proportion of the tangent of the angle complementary to angle BAC [IS THIS CORRECT?] to the secant of the angle that is complementary to angle BAC [IS THIS CORRECT?] is equivalent to the proportion of side AB to side AC.
Section 178 edit
The fifth method: Angle BAC and side BC are known, as was seen in the previous section, only instead of employing the procedure against angle BAC it is employed with its complementary portion, angle ABC. In this case all that was done with the then complementary angle BAC is now done with angle BAC.
Section 179 edit
The sixth method: If angle BAC and side AC are known, the proportion of the secant of angle BAC to the maximum sine is equivalent to the proportion of side AC to side AB, and the proportion of the secant of angle BAC to the tangent of angle BAC is equivalent to the proportion of side AC to side CB. Alternatively, the proportion of the secant of the complementary angle of angle BAC to the tangent of the complementary angle of angle BAC is equivalent to the proportion of side AC to side AB, and the proportion of the tangent of the complementary angle of angle BAC to the maximum sine is equivalent to the proportion of side AC to side BC.
In all these various methods there is no need to calculate the measure of the third angle, as was seen in a previous section [section 118].
Section 180 edit
If triangle under discussion is not a right triangle, whether it is equilateral, isosceles or neither - the method to use has been shown in a previous section [121], where the vertical side is drawn within triangle or outside it. This enables one to calculate the quantities of the right triangles created via use of the tangents and secants.
Section 181 edit
[To explain:] If the right angle line is drawn within the triangle, the proportions of the tangents of the intersecting angles are equivalent to the proportion of the sides opposite them. For example, in this figure:
The proportion of the tangent of angle BAD to the tangent of angle DAC is equivalent to the proportion of side BD to side DC. The proof of this is that the proportion of the maximum sine to side AD is equivalent to the proportion of the tangent of angle BAD to side BD, as was written in section 173. Similarly, the proportion of the maximum sine to side BD is equivalent to the proportion of the tangent of angle DAC to side DC. This being so, it follows that the proportion of the tangents of the angles to these angles' opposite sides are also equivalent, as was written in section [36, per commentary].
Section 182 edit
If the vertical line falls outside the triangle, as in this figure:
Then as is the proportion of the tangent of angle BAC to the tangent of angle CAD, so is the proportion of side BC to side CD. The proof of this is that the proportion of the maximum sine to side AD is equivalent to the proportion of the tangent of angle BAD to side BD. In the same fashion, the [proportion of the maximum sine to side AD is equivalent to] the proportion of the tangent of angle CAD to side CD. Since this is so, the proportion of the tangents of these angles to sides opposite these angles are also equivalent.
Section 183 edit
[Relating to section 181,] To continue, the proportion of the tangents of the angles opposite the vertical side are equivalent to the proportion of the sides that serve as secants. For example, as is the proportion of the tangent of angle ABC [to the tangent of angle ACD] so is the proportion of side DC to side BD. The proof to this is that the proportion of side CD to the maximum sine is equivalent to the proportion of side AD to the tangent of angle ABC, as was written in section 173. Similarly, the proportion of side DC to the maximum sine is equivalent to the proportion of side AD to the tangent of angle ACD. Now realize that in both of these proportions the middle terms are equivalent, and if so, the two extreme terms are proportional to each other as well, as has been shown in section [36].
Section 184 edit
If the vertical falls outside of the triangle, as was depicted in section 182, there is also of the equivalence shared by the proportion of the tangent of angle ABC to the tangent of angle ACB [ACD, according to commentary] and the proportion of side DC to side BD. The proof of this can be seen in the previous section as well.
Section 185 edit
The proportion of the secant of angle BAD to the secant of angle DAC is equivalent to the proportion of side AB to side AC. The proof of this is that the proportion of the maximum sine to side AD is equivalent to the proportion of the secant of angle BAD to side [AB, per commentary]. So too is the relationship of the secant of angle DAC to side AC. Therefore, these two secants themselves share an equivalency.
Section 186 edit
Continuing with the scenario where the vertical falls outside the triangle, there is equivalence between the proportion of the secant of angle BAD to the secant of angle DAC and the proportion of side AB to side AC. The proof to this is also based on the previous section.
section 187 edit
In addition, [based on section 181,] the proportion of the tangent of angle ABC to angle ACB is equivalent to the proportion of the tangent of angle DAC to the tangent of angle BAD, for just as the proportion of side DC to side BD so is the proportion of the tangent of angle ABC to angle ACB, as is written in section [183, per commentary]. The proportion of the tangent of angle DAC to the tangent of angle BAD [is equivalent to the proportion of side DC to side BD, as was written in section 181], if so [once it is seen that DC and BD are equivalent through the proportions the remainders are also equivalent. That means tangent ABC is to tangent ACD or tangent ACB as tangent DAC is to tangent BAD].
When the vertical falls outside the triangle there is another equivalence, and that is the proportion of the tangent of angle ABC to the tangent of angle [ACD, per commentary] and the proportion of the tangent of angle DAC to the tangent of angle BAD. The proof is based on what was written in section [184, per commentary].
Section 188 edit
And now, concerning a triangle that is not a right triangle, one can attain these measurements by different means: If two of the angles and the side between them are known, a right angle may be drawn from the third angle until the side that is known, like so,
where angles ABC, ACB, and side BC are known, and vertical side AD is drawn. Here the proportion of the tangent of angle ABC to the tangent of angle ACB is equivalent to the proportion of side DC to side BD, as was written in section [183, per commentary]. Through this, the three quantities may be known, [as angle ABD may be known, angle ADB is a right angle, and side BD is known as well. The same goes for triangle ADC, and the remaining quantities can also be known, as was written in section 110]. A similar procedure may also be used when the right triangle falls outside the triangle, following what was written in section [184, per commentary].
Section 189 edit
Alternatively, if two sides and the angle between them are known - for example when sides AB, AC, and angle BAC are known, and right angle AD is drawn: The proportion of side AB to side AC is equivalent to the proportion of the secant of angle BAD to the secant of angle DAC [as in section 185]. Through this method, [in triangle ADB angle ADB is a right angle, angle BAD is also known once the proportion of its secant to the secant of DAC is known, and then angle BAC is already known. Since this is so, the two angles that are divided by it are also known. Side AB is known. The same is also true in triangle ADC.]
If the right angle falls outside the triangle, the process would be according to section [186, per commentary].