Scheme II.

Analo-S1.As || AB:=R: : || BC:=seB, Ang. at Base gies 22.As=R: || AB: =tB: || AC, the Perpen.

Cafe VI. Scheme I.

Analo-S1.As || BC:=R::|| AC:-sB, Ang. at Bafe gies 2.As R: || BC::=sC: || AB, the Base.

Scheme III.

Analo-S1.As || AC:=R: : || BC :=seC, Ang. at Per. gies 2.As R: || AtC: || AB, the Base.

Thus I hope I have made the whole Affair of Trigonometrical Calculation facile and ready by the Sector, however the Proportions are Conftituted of Numbers and Sines, Numbers and Tangents, or Numbers and Secants; fo that though a Perfon come ever fo unskill'd in the Matter, I think when he has duely read and digested the Doctrine of this Chapter, 'tis fufficient to make him perfect, and dexterous in applying this Noble Inftrument to all the Purposes here treated of.

Note, I would advise the young Student, when he buyes a Sector, to have one 2 Feet in Length when opened ftrait; for then the Lines will be 12 Inches cach, and the Divifions thereof will be larger and clearer; however one of 18 Inches may do pretty well for common Ufes. Obferve alfo, that the Lines of Secants and fmall Tangents begin at an equal Diftance from the Center; for elfe fuch Analogies as have both Tangents and Secants in them cannot be performed thereby.

CHAP

Of the Sixth Method of Solving Rightangled Plin Triangles by Geometrical Conftruction.

Y Geometrical Construction is meant a Delineation of the Triangle in Lines by a Scale of Equal Parts and a Line of Chords, by the ftrict Method of Geometry taught in the Chapter of Geometrical Problems; fo that when the whole Triangle is thus projected in Plano, the Parts thereof unknown may be easily measured by the Scale from which the known or given Parts were delineated; that is, the unknown Sides measured on the Line of Equal Parts, and the Quantity of the unknown Angles from the Line of Chords; and thus the whole Triangle becomes known as foon as constructed.

The Scale for this Purpose ought to have a Line of Equal Parts divided Diagonally, fuch as is com mon on the Plain Scale; however a Line of Equal Parts of fome kind or other there muft of Neceffity be; and alfo a Line of Chords, or, in lieu thereof, an exact graduated Limb of a Quadrant. Whence this Geometrical Conftruction or Protraction of a Triangle may be performed by divers Inftruments, viz. all fuch as have on them the Lines aforefaid, as the common Plain Scale; The Sector; The Protractor; Sutton's Quadrant ; &c. But of all these the most ufual and useful are, the Plain Scale and Sector; the

first having a Line diagonally divided, which gives an Answer much nearer the Truth, than one that is not thus divided can do; and the Latter, as it is capable of being fet to any Radius, which is a Property peculiar to its felf.

The Geometrical Conftruction of the Six Cafes of Right-angled Plain Triangles, according to what is given in each, here follows.

First, Draw the Bafe Line BA at pleasure, and from the Diagonal Line on the Plain Scale, or Line of Equal Parts on the Sector, take the Length of the Base 230, with your Compaffes, and fet it from B to A.

C

Secondly, On A raife a Perpendicular (by Problem II.) indefinitely continued.

Thirdly, Take the Chord of 60 Degrees from the Line of Chords on the Plain Scale, off from the Limb of the Protractor or Quadrant; or laftly, from the Parallel Chords of 60o on the Sector (conveniently opened) and therewith (as a Radius by Theorem XVII.) fetting one Foot of the Compaffes in B, ftrike the Arch dc; and from the faine Line of Chords take the Quantity of Angle B, viz, 36° 30', and fet from c to d.

Fourthly, From B draw a Line through the Point d 'till it meet the Perpendicular in C. Then is ABC the Triangle required.

Fifthly, Measure the Perpendicular AC on the fame Line or Scale you took AB from, and it will be found to be 170.19.

Sixthly, Measure alfo on the fame Line the Hypothenufe BC, and you will find it to be 286.12; and thus the whole Triangle is folved.

Cafe II. Given the Perpendicular and Angles.

This Cafe is conftructed in the fame Manner as the laft, if you proceed with the Perpendicular AC, and the Angle C, as I before directed for the Bafe AB and Angle B.

Scheme omitted.

Cafe

First, Draw a blank Bafe Line BA with one Foot of the Compaffes, on which at B make an Angle, as directed in Cafe I.

Secondly, Through the Point d draw the Line BC indefinitely, and from a Scale of Equal Parts take 286.12 and fet from B to C thereon.

Thirdly, From the Point C let fall a Perpendicular (by Problem III.) to the blank Line BA, cutting it at Right Angles in the Point A, and join A, B for the Bafe.

Fourthly, Meafure the Bafe BA and the Perpendicular AC, on the Scale, and you will find the first to be 230; and the other 170.19, as before.