iii THE PREFACE I N the Course of my Studies on this Branch of the larity and Deficiency in the Writings of most 44thors on this Subject. I have therefore in the following Treatise endeavour'd to redress this Grievance, and to render this Science more plain, easy, and compreheile five ; to which End I have laid down a large Number of Definitions, Theorems, and Problems, which are absolutely necesary to be learned. The Definitions inform us of the Nature, Names, and Différentes of the several Lines and Figures usedikereine , and also other Terms of the Art. The Theorems contain all the Mystery of tbe Science, on a Demonflration, and due understanding of which depends the Rationale of Trigonometry; that is, the Reasov of all the various Operations of the Art ; Thirty-one of the best Theorems I have collated for this Purpose, which serve more immediately for understanding the Reason of the Methods for making the Canon of Natural Sínes, Tangents, and Secants. I be Theorems ought to be well studied. In demosArating them I have taken a new Method, after the Manner of Mr. Le Clerc, in bis Practical Geometry, which I think is the most clear, perfpicuous and easy of all others. The Problems are adapted to the Capacity of all, enabling them to make a Geometrical ConftrucA 2 tion tion of any Process or Operation in Trigonometry; in which the Learner ought to be very perfeet. In the next Place follow the several Methods of constructing a Canon of Natural Sines, Tangents and Secants, invented by the Learned, and here illustrated and exemplified in Numbers. This Matter bowever neglected in Books of Trigonometry, is of the last Importance to finish an Artist in this part of Learning: Having learni'd this, the Reason of the Logarithmetic Canon is easy, as will appear in due Place. After having thus laid the Foundation Principles, I proceed to the Method of solving all the Six Cases of a Right-angled Triangle, in all the three Varieties, and that by ten seves ral Methods; some by Natural Numbers, fome by Artificial Numbers, and others by Instruments. I have fewn how the Method by Logarithms depends on these by Natural Numbers; and how the Metkod:13. Inftruments depends on both; I let the young Leariier at once into e be. Myftery, or Theory and Practice of each:Yethods and bave taken care to teach him nothing, from the which he may not be able to render a Reason. Theriforė:F.Have contriv'd a new Method to contince bin of the Retinale of every Analogy, by Logarithms; or why we say Sine, or Tangent, or Secant; and why the Parts of Analogy are placed as they are, and the Reafon of the large Indices to those Sort of Numbers; and have endeavour'd to make the whole clear ard'intelligible to the Learner by vew Schemes for that Purpose. Ilute also describ'd the Nature and Use of the several Lines on the Instruments; as the Sector, Plain Scale, Sliding-Rule, &c. as I have gone along; and have il lustrated tkeir Uses in the Dočtrine of Triangles. You kare also an Ample Collection of Anomalous Cases of a Righe Triangle, with their Solution by Algebraic The Oicnis. Tben Then follows tbe Method of folving all the cases of Oblique Triangles, by the foresaid new Method, and Schemes ; sewing the Rationale of every Process. And in the last Place you have a Variety of Theorems, new and curious, for finding the Area of any Plain Triangle, by having given any of the Sides, or Sides and Angles together. In the second Part of this Work, I have applied the Doctrine of Plain Trigonometry to the ten Matbema. tical Arts and Sciences mention'd in the Title-Page. In each of which I have given a General Account of the Art, its Principles and Maxims, the Nature and Pro-. perties of its several Parts, and the Use of Trigonometry in the whole : By which means the Reader will meet with a Kind of Epitome of the Mathematics, so far as it depends on Plain Trigonometry: Here are inserted a great Number of very rare and curious Propositions in Cosmography, Geographiy, Aftrcpomy, Gunnery, Mechanics, Surveying, Optics and Perfpective, which are not to be found in any one Auther (that I know of) on this Subject, and yet such as properly appertain thereunto, the Reader may see the Particulars in the Table of Contents. One Thing I would acquaint the Reader with, that I have been concise, yet I think sufficiently prolix, in those Parts which are in every Book that treats of this Art, viz. Navigation, Altimetry, Fortification, and in some other Parts have perform’d the Operations by the Inftrumental Methods, as by the Gunter, the Scctor, &c. and therefore in such cases, the Reader is noi to expect fuch Matkematical Exactress as by Logarithms, because the Matter does not require it. Upon the whole, I have onnitted nothing that I could think of which might render this IVork a compleat Guide to the young 7 rigonometer; and I hope in the Uje it will be found to be fich. As for what belongs to Spherical Trigonometry To conclude, having given a general Account of this Piece, its Occafion, Subject and Method; I can not but fit in the perufal of it, which was intended in its Com- posure by bim, who will never think much of any Pains |