plane and spherical triangles; the whole being fully and clearly explained. This is the first work in which the tangents and secants are carried to 7 places of decimals to the last degree of the quadrant. A complete and masterly work on Trigonometry by Pitiscus, was published at Frankfort, in 1599; the triangular canon is here given, and its construction and use clearly described, together with the application of Trigonometry to problems of surveying, altimetry, architecture, geography, dialling, and astronomy; forming the most commodious and useful treatise on the subject at that time extant." Several other writers on Trigonometry appeared towards the close of the 16th, and at the beginning of the 17th century, of whom Christopher Clavius, a Jesuit of Bamberg, may be considered as one of the chief. In the first volume of his works, (which were printed at Mentz, in 5 volumes, folio, 1612,) he has given an ample and circumstantial treatise on Trigonometry. In this work the canon of sines, tangents, and secants, is computed for every minute to 7 places of decimals, and carried forward to the end of the quadrant, the sines having their differences computed to every second, and construction of the tables being accompanied with clear and satisfactory explanations, chiefly derived from the methods of Ptolemy, Purbach, and Regiomontanus. Van Ceulen, in his celebrated treatise De Circulo et adscriptis, first published about the year 1600, treats of the chords, sines, and other lines connected with the circle; which work, with some other of Van Ceulen's pieces, was afterwards translated into Latin, and published at Leyden, in 1619, by Willebrord Snellius, who has also himself given in his Doctrina Triangulorum Canonica, the construction of sines, tangents, and secants, together with a very useful synopsis of the calculation of plane and spherical triangles. A canon of sines, tangents, and secants, for every minute of the quadrant, was published in 1627, at Amsterdam, by Francis Van Schooten, the ingenious commentator on the Geometry of Des Cartes. His assertion, that his table was without a single error, has been since found to be incorrect; some of his numbers have been discovered to err in the last figure, being not always calculated to the nearest unit. & In the early ages of Geometry the circumference of the circle was divided into 360 degrees, each degree into 60 minutes, each minute into 60 seconds, &c.; this method was adopted by the moderns, and still prevails among the English, and most other nations in Europe; but the French mathematicians have introduced an improvement, which, when it is generally understood and adopted, will be of the greatest advantage to Trigonometry. Towards the latter part of the eighteenth century, a new system of weights and measures was instituted in France, in which they were decimally divided and subdivided; this was followed by another of equal importance, a new division of the quadrant. By this new method, the whole circumference is divided into 400 equal parts called degrees, each degree into 100 minutes, each minute into 100 seconds, &c. consequently the quadrant will contain 100 degrees. One advantage in this method is its convenient identity with the common decimal scale of numbers, for 1°, 23′, 45′′, in the new French scale will be expressed by the very same figures in common decimals, viz. by 1.2345o; in like manner 21o, 3′, 4′′, French, is expressed by 21.0304o common decimals; 170o, 1′, 2′′, 34" by 170.010234°; 5′, 0′′, 11" by .050011; 12′, 18", 14′′ by .121314, &c. Among the works on this plan at present in use, are Les Tables Portatives de Callet, 2 Edit. Paris, 1795; the Trigonometrical Tables of Borda, improved by Delambre, 4to. an IX. ; and the tables lately published by Hobert and Ideler, at Berlin. Likewise tables on the above plan, to an extent hitherto unknown, have been for many years under the hands of M. Prony, assisted by a number of able mathematicians, a work which, besides its great usefulness, will be the most ample monument existing, of human industry in the province of calculation. To reduce degrees, minutes, &c. of the French scale, into degrees, minutes, &c. of the common scale, and vice versá. Since the quadrant is divided by the French method into 100o, and by the common method into 90°, . 100° French =90° common; To reduce French degrees, minutes, &c. into common, RULE. Express the French measure decimally, subtract from this To of itself; mark off the proper decimals in the remainder, multiply these by 60, mark off the decimals; multiply these again by 60, and mark off the decimals as before, &c.; the resulting whole numbers will be the degrees, minutes, seconds, &c. required, according to the common scale. EXAMPLES.-1. In 34°, 56′, 32" French, how many degrees, minutes, seconds, &c. common? The invention of logarithms by Lord Napier, in 1614, and their subsequent improvement by Mr. Henry Briggs, greatly facilitated the practical operations of Trigonometry. Besides the invention of logarithms, we are indebted to Napier for the method of computing spherical triangles by means of the five circular parts, and other valuable improvements in spherical Trigonometry. The doctrine of infinite series, introduced about the year 1668, by Nicholas Mercator, and improved by Newton, Leibnitz, the Bernoullis, and others, soon found its application to Trigonometry, by furnishing expressions for the sines, tangents, &c. for which purpose the exponential formulæ of Mr. Demoivre are extremely convenient. But the greatest and most useful improvement of modern times in the analysis of sines, co-sines, tangents, &c. which Therefore 34°, 56′, 32′′ French =31°, 6′, 24′′, 46′′, 08 common. 2. In 8o, 12′, 3′′ French, how many degrees, minutes, &c, common? ARs. 7°, 18′, 38", 31". 3. In 12o, 1′, 2′′ French, how manydegrees, &c. common? 4. In 9°, 8', 7" French, how many degrees, &c. common? To reduce common degrees into French. RULE. Turn the minutes, seconds, &c. into decimals, to the whole add of itself; then the integers of the sum will be degrees, the two left hand decimals minutes, the two next decimals seconds, &c. EXAMPLES.-1. To reduce 34°, 56′, 32′′ common, to French measure. Add of the same = 3.882469 The sum is 38.824691=38°, 82′, 46′′, 91′′ French. 2. In 24°, 44′, 6" common, how many degrees French? Ans. 24°, 15'. 3. Turn 23°, 27′, 58" common into French. Ans. 26o, 17′, 35′′. 4. Turn 1o, 2', 34" common into French. we owe to the penetrating, comprehensive, and indefatigable mind of the venerable Euler; by substituting the analytical mode of notation, in the room of the geometrical, which had hitherto been chiefly used, he simplified the methods of preceding writers, investigated a great variety of formulæ, applicable to the most difficult cases, and made the trigonometrical analysis assume the form of a new and interesting science. Admitting that the Continental mathematicians are our superiors in the theory of Trigonometry, as well as in their writings on the science, still we have some very good and useful treatises on the subject; the chief of which are those of Thomas Simpson, Emerson, Maseres, Horsley, Keith, Vince, and Woodhouse; but Mr. Bonnycastle's Treatise on Plane and Spherical Trigonometry, is the most complete work on the subject of any that have hitherto appeared in this country. * See the Quarterly Review for November, 1810, page 401. |