before we have met with in Right-angled Triangles, from all the Authors on the Subject, that I have yet feen, Yea, even the great Mathematical Critic, Mr. Cunn himself, has taken no notice of this Matter; when he undertook to correct Dr. Keil, Dr. Harris, Mr. Cafwell, Mr. Heynes, &c. in the Affair of Ambiguities; (which he hath performed with laudable Succefs.) This happens to be the Effect of a wrong Notion of what is properly a Cafe of the Solution of a Triangle, which I have more than once fhewn to be, when there are fufficient Data, or Parts given. whereby the whole Triangle may be refolved or become known in every Part, and which require different Analogies for the Solution. The Writers on this Subject generally make twelve Cafes of Oblique Triangles, but I a little wonder at their Moderation; for according to their Notion of a Cafe, they might very reafonably have claimed eighteen; for that was the leaft Number that could be denied them. But 'tis moft certain that inftead of twelve, there are really but fix Cafes of Oblique Triangles; for 1. There may be two Angles given, and a Side oppofite to one. 2. There may be given two Angles, and the included Side. 3. There may be given two Sides, and the Angle included. 4. There may be given two Sides, and an oppofite Angle. 5. There may be given all the three Sides; and 6. There may be given all the three Angles. And these are all the Varieties of Cafes which can happen in an Oblique Spherical Triangle; for it matters not what two Sides, or what two Angles, are given in in any Cafe; the Rule or Analogies being ftill the fame. Having thus fettled and determined the Number of Cafes in Oblique Spherical Triangles, as I have before done in all other kinds of Triangles; and which indeed was neceffary to be done, fince according to the juft Obfervation of the Poet; Eft Modus in Rebus, funt certi denique Fines, I fhall now treat of the Method of folving these forts of Triangles by means of a Perpendicular, which being let fall, reduceth the Oblique Triangle into two Right-angled ones; and this (as Mr. Cunn obferves) is undoubtedly the shortest and best Method for finding the Quæfita in any Cafe where the Solution cannot be obtained at one Operation. In order to this, we must firft determine whether the Perpendicular falls within or without the Triangle given; which Matter is fometimes certain, and fometimes ambiguous or doubtfull. If the Angles at the Bafe (or Side on which the Perpendicular falls) be both known, then 'tis certain how the Perpendicular will fall; because if they are both of a Species, that is, both Acute or both Obtufe, then will the Perpendicular fall within the Triangle, by Theorem 22; but if they are of a different Species or Affection, that is, one Acute, and the other Obtufe, then fhall the Perpendicular fall without the Triangle. If only one of thofe Angles be known, 'twill be uncertain whether the Perpendicular will fall within or without the Triangle ; all this is evident from the Demonftration of the aforefaid Theorem 22. To illuftrate the Matter, I have prefented the Reader with three Diagrams in the following Synopfis, which prefents to the View all the whole Myftery of Oblique Trigonometry; for in the Triangle BCD, if the Side B D be fuppofed to be the Bafe, as in Diagram 1, then because the Angles at the Base, B, D, are of the fame Affection, viz. both acute, therefore the Perpendicular C A falls within the Triangle, and divides it into two Right-angled Triangles, A B C, and AD C. But in Diagram 2, BD being made Bafe as before, because the Angles B and D at the Bafe, are of a different Affection, therefore the Perpendicular AC falls without the Triangle and makes two Right-angled Triangles ABC and AD C, by Addition of the Latter. The fame thing happens in Diagram 3, in an inverfe Order to the laft. Thus in Diagram 2, the Triangle BDC having two acute Angles B, C, at the Bafe B C, is divided by the Perpendicular D falling within, into the two Right-angled Triangles A B D, a CD; but in Diagram 1, and 3, the Perpendicular falls without, becaufe B and C are of different Affection. So likewife in Diagram 3, the Triangle CBD Having the two acute Angles C, D, at the Bafe CD, is the Perpendicular B falling within, divided into the two Right-angled Spherical Triangles CB, DB; but in Diagram 1, and 2, the Perpendicular falls without, because C and D are of a different fort. Moreover becaufe either Side of a Triangle may be made the Bafe, and from either Angle we may let fall a Perpendicular; therefore it is that there is fuch variety of Triangles produced as is obfervable in the three Diagrams, viz. three in each Diagram, two Right-angled, and one Oblique-angled. Z VOL. II. Thus |
