CHAP. XIII.

Of the Seventh and Eighth Methods of Solving Right-angled Plain Triangles by the Practical Trigon, and Sinical Quadrant.

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HE two Inftruments, whose Uses are here to be described, are the most expeditious in Practice of any by Natural Lines; yea, they are in themfelves the most natural of any Means; for either at once constitute the Triangle, and fhews the Dimenfions of every Part thereof, by Infpection only, in Natural Numbers on the respective Lines. But be cause those Inftruments are very fcarce, and in few Perfons Hands; I have therefore given the Young Geometer a Figure both of the Trigon and Senical Quadrant, and their Defcriptions which here follow.

A Defcription of the Trigon. Fig. I.

The Trigon is an Inftrument confifting of three, and fometimes of four, Parts or Pieces: The firft is the fixed Piece BA, which reprefents the Bafe of a Right-angled Triangle, and is graduated into an hundred Equal Parts. The fecond Piece is BC, and is inferted into BA by the Joint B, on which it moves, and may be fet to a given Angle with BA; this Part reprefents the Hypothenufe of a Triangle. The

third

third Piece is AC, and is made to move backward and forward on BA, by means of the Socket S, fo as to be in a perpendicular Pofition thereto; and therefore this Part reprefents the Perpendicular or Cathetus of the Triangle, and is alfo graduated into an hundred Equal Parts. The fourth Part is the Semiquadrantal Arch D, divided into 45 Degrees, being fixed in the End of the Part BA, in order that the Piece BC may move commodioufly by it, and be fet to the given Quantity of any Angle under 45° thereon. By this Defcription, and even from the very view of the Figure it felf, 'tis eafy to conceive how very naturally this Inftrument at once both forms and fhews the Quantity of each Part of the Triangle unknown. The Reason why the Arch is here described as containing only 45°, and no more, will appear in the Ufes hereof by and by. And here 'tis to be obferved, that because this Inftrument may be opened to an Angle of 45°, and each Leg in fuch Cafe may contain 100 Parts, 'tis therefore neceffary the Hypothenufal Part BC, fhould be divided into 141.4 of fuch Parts as being the Square Root of 20000, the Sum of the Squares of the Legs. See Theorem XI.

Note, If the Leg AC were made to move on the Socket S circularly, this Inftrument might also be ufed in folving Oblique-angled Triangles.

A Defcription of the Sinical Quadrant.

This Quadrant hath, like all others, a graduated Limb of 90 Degrees; and its two Rectilineal Sides (or Radius's) BD and BE, divided into 100 Equal Parts, from each of which are drawn Right Lines to the Circular Limb, mutually interfecting each other on the Superficies of the Quadrant, and make the Sines and Co-Sincs of as many Divifions in the Quadrantal

Quadranta! Limb. On the Center of the Quadrant B is fixed a moveable Index or Label BC, graduated alfo into 100 Equal Parts; this Index thus moving on the Quadrant is to be fet to any Given Angie, and ferves for the Hypothenufe in any Triangle, the other two Legs which make the Bafe and Perpendicular being thofe Lines, or Parts thereof which arife from either graduated Side and meet the graduated Edge of the Label, and the Side of the Quadrant it felf; all this is evident by a View of the Figure II, only.

It is not my Purpose (here at least) to fhew the other Ufes that may be made of the Quadrant here described, and the Trigon, but only that of folving Triangles thereby; and that in the Trigon is as follows.

The Ufe of the Trigon.

Admit there be a Right-angled Triangle, in which there is Given the Bafe BA=82, or 820; and the Angle at Bafe B=30° 00'; and its Complement of Course, C=60° oo' required the Hypothenufe BC, and Perpendicular AC.

Direction I. Set the Hypothenufal Part BC to 30° oo' on the Limb, then flide the Perpendicular Part AC to 82 (or 820) on the Base Part BA; this being done, the Triangle is formed; and from the Center B to the Common Interfection at C, is contained 94.68 (or 946.8) =BC, the Hypothenufe; and from A to C, is intercepted 47.34 (or 473,4) AC, the Perpendicular; thus with Eafe, and in a Moment is fuch a Triangle folved by Infpeétion only.

Direction II. If the Angle at Base, in the aforefaid Cafe, be Given greater than 45°; the moveable Hy

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pothenufe

Pothenuse must be fet to the Complemental Angle, and the Inftrument being inverted, viz. the Perpendicular Piece made Bafe, the fmall Divifions muft be used, as before the larger were, and the Anfwer will appear on the other Parts as before; but not exact enough for any confiderable Purpose, unless the two Legs are at least a Foot in Length each; and fuch as chufe this Inftrument, may as well have it 2 or 3 Feet long, as one; and their Work will be portionably more exact.

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Direction III. When both the Legs are Given, do thus; Set the Perpendicular Leg AC to what is Given on the Bafe Leg BA, then move the Hypothenufal Part BC to the Given Parts on AC; and thus is the Triangle formed, and the Solution evident by Infpection; for the Point of the Hypothenufe on the Limb D'fhews the Quantity of the Angle at Bafe B, and fo of its Complement C; and the Parts intercepted between B and C on the Hypothenufe, are its required Length.

Direction IV. When Bafe and Hypothenufe are Given; flide the Perpendicular Part AC to the Given Part of the Base BA, then move the Hypothenusal Part up or down, 'till the Given Part thereon meet the Perpendicular AC; and thus the Triangle is formed, and the Angles and other Side, are known by Infpection.

Direction V. When the Hypothenuse and Perpendicular are Civen; move the Parts BC and AC, fo together, that the Given Parts on both may coincide in C; when this is done the Triangle is formed, and folved by Inspection.

Direction

Direction VI. When the Hypothenufe and Angles are Given; Set the Hypothenufal Part BC to the Quantity of the Angle B on the graduated Limb D, then move the Perpendicular Part AC to the Given Part on BC; and thus the Triangle is formed, fhewing the Quantity of the other two Sides BA and AC.

Direction VII. When the Perpendicular and Angles are Given; Set the Hypothenufe BC to the Degrees of B on the Limb D, then flide the Perpendicular backwards and forwards, 'till its Given Parts meet the Hypothenufe in C; fo is the Triangle formed, and the Sides visible on their respective Parts BC and BA.

Thus have I given Directions for the Ufe of this Practical Inftrument in all Variety of Cafes. I proceed now to

The Ufe of the Sinical Quadrant.

In the following Directions, I call thofe Lines which are drawn from the Divifions of the Side of the Quadrant BD upwards, Right Sines; and thofe which are drawn from the Side BE, across the Quadrant, Tranfverfe Sines.

I. Let there be Given the Bafe BA=63, and the Angle B 38° 30'; its Complement being 51° 30′; to find the Sides BC and AC, by the Sinical Quadrant.

Direction I. Set the Index to the Angle B on the Limb; then obferve where the Right Sine of 63 Parts on BD interfects the Index, and you'll find it to be in C in the Divifion of 80.5, which therefore is the Length of the Hypothenufe BC on the Index. Next for the Perpendicular AC, obferve where the

Point