From what I have here faid concerning this Cafe I. 'twill be very easy, in the fame Manner, to folve any of the other following Cafes; which therefore I leave to the young Learner's Exercife and Diverfion.

Hence 'tis manifeft this most excellent Art might be much more generally understood, and ufed not only by Scholars, but by every common Trader, Artificer or Husband Man, by the cheap and easy Means of a Table of Natural Sines, Tangents and Secants, and a common Sliding Rule.

And I am ftrangely surprised at the fupine and ftupid Indolency of many young Perfons, who have (and might have by good Husbandry) Time enough on their Hands, Money enough in their Pockets, and Intellects enough in their Heads, yet notwithstanding this, will expend neither in the Study of this, or any other Noble Art, or Part of Mathematical Learning; which would in fo moft agreeable a Manner, enoble their Nature; enrich their Minds; and elevate them above, and rescue their Reputations from the rude and barbarous Vulgar. Inftead of which they idly chufe to bury their Talents, and wretchedly live, and ignominously die without Remembrance.

Of the Fourth Method of Solving Rightangled Plain Triangles, by Gunter's Scale and Compaffes.

HIS celebrated Line of Artificial Numbers, (whofe Construction and its Ufe on the Sliding Rule, were fhewn in the laft Chapter) first received

its

its Name and Being, from the famous Profeffor of Geometry at Gresham College, Mr Gunter; of whom it has been commonly called Gunter's Line, Gunter's Scale, or fimply, The Gunter: But what I here call Gunter's Scale, is a Line of Artificial Numbers, Sines and Tangents, laid down on a Plain Scale or Rule, and are unmoveable.

On this Scale, the fame Things are performed by the Compaffes, as on the Sliding Scale or Rule, by the Sliding Piece, and the whole Method depends on this cafy

General Rule.

Set one Foot of the Compaffes in the firft Term of the Analogy, (be it Number, Sine or Tangent,) and extend the other Foot (to the Right or Left) 'till it fall on the Term of the Analogy that is with the fame Kind with it felf (whether it be the Second or Third); and that Extent of the Compaffes will reach from the remaining Term) the fame Way as before) to the fourth Term or Answer.

The fame Things, or the fame Schemes and Analogies are to be here used, as before, in folving the firft Cafe.

Cafe I. Scheme I. To find the Perpendicular AC.

Set one Foot of the Compaffes in the Sine of 53° 30', and extend the other to the Sine of 36° 30'; then that Extent will reach from 230 to 170.19, in the Line of Numbers, and is the Anfwer.

To find the Hypothenufe BC.

Set one Foot in the Sine of 36° 30', and extend the other to Radius or 90°; then that Extent will reach from 170.19 to 286.12, in the Line of Numbers for Anfwer.

Cafe

Cafe I. Scheme II. To find the Perpendicular AC.

Set one Foot of the Compaffes in the Tangent Radius or 45°, and extend the other to the Tangent of 36° 30'; that Extent in the Line of Numbers will reach from 230 to 170.19=AC, as before.

To find the Hypothenufe BC.

Note, That because (as I have faid before) there is no Line of Artificial Secants, and this Analogy containing a Secant, it can only be folved by the Line of Numbers with the Compaffes here, as with the fame Line and Slider in the foregoing Chapter, from the Natural Numbers in the Table. Therefore fet one Foot in Radius 1000, and extend the other to 230; then that Extent will reach from 1244 to 286.12= BC, as required.

Cafe I. Scheme III. To find the Perpend. AC.

Set one Foot of the Compaffes in the Tangent of 53° 30', and extend the other to Radius or 45; then in the Line of Numbers that Extent will reach from 230 to 170.19=AC, as before.

To find the Hypothenuse BC.

Becaufe this Analogy is here alfo by Secants, therefore by the Line of Numbers, from the Natural Numbers of the Table, do as before taught; Set 1000 to 170.19; that Extent will reach from 1681 to 286.12 BC, the Side fought.

Cafe

Cafe IV. Scheme II. To find the Angle B.

In the Line of Numbers, fet one Foot in 230, and extend the other to 170.19; then in the Tangent Line, that Extent will reach from Radius or 45°, to 36° 30', which is the Tangent of the Angle B.

Note, The Side BC is to be found by the Line of Numbers, &c. as directed in the Firft Cafe, Scheme II. III.

Cafe IV. Scheme III. To find the Angle C.

In the Line of Numbers, Set one Foot in 170.19, and extend the other to 230; then will that Extent reach from Radius or Tangent of 45°, to the Tangent of 53° 30', the Quantity of the Angle C.

The Learner that has regularly come thus far, need not, I fuppofe, be told that though the Tangent of any Angle, and of that Angle's Complement be in the fame Point of the Line, as in this Cafe, 53° 30' and 36° 30' are; yet he may eafily know which is the Angle required by the Second Term of the Analogy; for if that be greater than the First, the greatest Tangent is the Angle required; as here 230 being greater than 170.19 makes it certain that 53° 30' is the Angle fought; and the Contrary.

The Reader is fuppofed in each of thefe Operations to have his Eye on the Analogies in the Synopfis, by which they are performed. And thefe being all the Varieties by the Gunter and Compaffes; I proceed to the Fith Method by the Sector.

CHAP.

CHA P. X.

Of the Fifth Method of Solving Rightangled Plain Triangles by the Sector.

B

Y this moft noble and most useful of all Mathematical Inftruments, The Sector, not only all the common Operations of the Mathematical Sciences are moft eafily and commodioufly performed; but particularly the whole Business of Trigonometrical Calculation, of every Kind, is hereby very eafy, expedite and perfect; For all manner of Analogies, whether by Numbers, Sines, Tangents, Secants, are alike

refolved here.

The Invention of this wondrous Inftrument is founded on the 4th Prop. of the 6th Book of Euclid, or Theor. XV beforegoing. Where 'tis demonftrated, that Parallels to the Bafe of any Plain Triangle, bare the fame Proportion to the Bafe, as the Parts of the Legs above the Parallel, do to the whole Legs. But more of the Nature and Conftruction of the Secor is to be learned from thofe who have wrote purposely thereon.

The Lines on the Sector are various, but I have to do with no more those which ferve more immediately to the Solutions of Triangles, and they are those which follow.

Firft, A Line of Chords on each Leg of the Sector; they proceed from the Center to the End of the Legs, where they terminate in a brass Point, at 60°, the Chord thereof being equal to Radius.

Secondly,