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THE Triangle which I shall make use of in the severall Cases belonging to a right-angled plain Triangle shall be this following, C A B, in which
| parts. | |||
| A B, the Base, | contains | 180 | |
| C A, the Perpendicular, | 135 | ||
| C B, the Hypotenuse, | 225 | ||
| And | deg. | m. | |
| A, the right Angle, | contains | 90 | 00 |
| C, the Angle at the Per. | 53 | 07 | |
| B, the Angle at the Base, | 36 | 53 |
DRaw a Line A B, and from your Scale of equal parts take 180, and set them from A to B; then on the Point A raise the Perpendicular A C, and because it contains 135, take 135 parts from your Scale of equal parts, and set them from A to C; then draw the Line C B: which three Lines will constitute the Triangle C A B.
Now to finde the Angles C and B, take in your Compasses 60 degr. of your Line of Chords, and setting one foot in C, describe the Arch f g; also setting one foot in B, describe the Arch d e: then take in your Compasses the distance from f to g, which measured upon your Line of Chords will reach from the beginning thereof to 53 degr. 7 min. and such is the quantity of the Angle at C.—In like manner take the distance between d and e in your Compasses, that distance applied to your Line of Chords will reach to 36 degr. 53 min. —Or, when you had found the quantity of the Angle C to be
53 degr. 7 min. if you had subtracted that from 180 degr. the remainder would have been 36 degr. 53 min. the quantity of the Angle at B, without drawing of the Arch d e, and measuring it upon your Chord.
For such as have a Canon of artificial Sines, Tangents and Logarithms, and would resolve this Case by them, this is
The Analogie or Proportion.
As the Logarithm of A B is to the Logarithm of A C,
So is the Radius to the Tangent of B.
DRaw a right Line A B containing 180 parts of your Scale of equal parts, also out of the same Scale of equal parts take 225, your Hypotenuse, and setting one foot of your Compasses in B, with the other describe the obscure Arch h k; then on the Point A raise the Perpendicular A C, which will cut the obscure Arch h k in C; then draw the Line C B, so have you the Triangle C A B: then may you measure the quantity of the Angles at C and B as in the last Case. And so will C be 53 degr. 7 min. and B 36 degr. 53 min.
The Analogie or Proportion is,
As the Log. of C B is to the Radius,
So is the Log. of the Side A B to the Sine of C.
DRaw a right Line A B containing 180 parts of your Scale, for the Base of your Triangle; then taking 60 degr. from your Line of Chords, on the Point B describe the Arch
d e, and (because the Angle at B contains 36 degr. 53 min.) take 36 degr. 53 min. from your Chord, and set it from d to e, and from B, through the Point e, draw the Line B C. Also upon the Point A erect the Perpendicular A C, crossing the Line B C in C. So have you formed the Triangle C A B. Lastly, take the length of the Line A C in your Compasses, and measuring it upon your Line of equal parts, you shall find it to contain 135. And that is the length of the Perpendicular C A.
The Analogie or Proportion is,
As the Sine of the Angle at C is to the Log. of A B,
So is the Sine of the Angle B to the Logar. of C A.
Or,
As the Radius is to the Logar. of A B,
So is the Tangent of B to the Logar. of C A.
DRaw a right Line C B containing 225 of your Line of equal parts; then taking 60 deg. out of your Line of Chords, set one foot of the Compasses in B, and with the other describe the Arch e d; also (the Compasses continuing at the same distance) place one foot in C, and with the other describe the
Arch g f. Then from the Point B, and through the Point d, draw a right Line; also from the Point C, and through the Point f, draw another right Line: these two Lines will intersect or cross each other in the Point A, forming the Triangle C A B. Lastly, take the Line A B in your Com∣passes, and applying it to your Scale of equal parts, you shall finde it to contain 180; and that is the length of the Base A B. Likewise A C being taken in the Compasses, and measured upon the Line of equal parts, will be found to contain 135, which is the length of the Perpendicular C A.
The Analogie or Proportion is,
As the Radius is to the Logarithm of C B,
So is the Sine of C to the Logarithm of A B,
And the Sine of B to the Logarithm of C A.
DRaw a right Line A B containing 180 of your Scale of equal parts, and upon the end A erect a Perpendicular A C. Then take out of your Scale of equal parts 225, (the length of your Hypotenuse given,) and setting one foot of the Compasses in B, with the other describe the Arch h k, cut∣ting the Perpendicular A C in C, then draw the Line C B: so have you constituted the Triangle C A B. Lastly, take in your Compasses the length of the Line A C, and apply it to your Line of equal parts, where you shall finde that it will contain 135: and that is the length of the Perpendicular C A.
The Analogie or Proportion is,
1. Operation.
As the Logarithm of C B is to the Radius,
So is the Logarithm of A B to the Sine of C.
2. Operation.
As the Radius is to the Logarithm of C B,
So is the Sine of B (the Complement of C) to the Log. of C A.
DRaw a right Line A B containing 180 parts of your Scale, for the Base of your Triangle, and on the end A erect a Perpendicular A C. Then take 60 degr. out of your Line of Chords, and upon the Point B, with that distance, describe the Arch d e; and (because the Angle at B is 36 degr. 53 min.) take 36 degr. 53 min. from your Line of Chords, and set it upon the Arch from d to e. Then from B, through the Point e, draw a right Line, till it meet with the Perpendicular before drawn, which it will do in the Point C. And thus have you protracted your Triangle C A B. Lastly, take in your Compasses the length of the Hypotenuse C B, and measure it upon your Scale of equal parts, and you shall finde it to contain 225.
The Analogie or Proportion is,
As the Sine of C is to the Logarithm of A B,
So is the Radius to the Logarithm of C B.
DRaw a right Line A B containing 180 equal parts, and upon the end A erect the Perpendicular A C, and out of your Scale of equal parts take the length thereof 135, which set from A to C, and draw the Line C B, which con∣stitutes
the Triangle C A B. Lastly, take the length of the Hypotenuse C B in your Compasses, and measuring it upon your Line of equal parts, you shall finde it to contain 225.
The Analogie or Proportion is,
1. Operation.
As the Logarithm of A B is to the Logarithm of C A,
So is the Radius to the Tangent of B.
2. Operation.
As the Sine of B is to the Logarithm of C A,
So is the Radius to the Logarithm of C B.
These are the severall Varieties or Cases that can at any time fall out in the Solution of Right-angled plain Triangles, wherefore we will now proceed to the Solution of Oblique plain Triangles.