T R I 



fnealh, feparate above, fome of them often abortive, and the 

 common (heath finally fplits into two ; anthers oblong. 

 Pyi. Germen fuperior, ovate, fmall ; flyle ihort, afcending ; 

 ftigma capitate, flat, with a membranous border. Perk. 

 Capfule oblong, acute, with three angles and three inter- 

 mediate channels, of one cell and three valves, which are 

 boat-fhaped, double, the outer coat coriaceous, inner mem- 

 branous, lined with wool. Seeds numerous, roundifli, en- 

 veloped in long wool, and connefted with a triple thread- 

 iliaped receptacle. 



Eff. Ch. Calyx in five deep fegments. Corolla papi- 

 lionaceous ; ftandard pitted at the bafe. Ne6lary two 

 fcales at the bafe of the germen. Some filaments imperfeft. 

 Capfule leguminous, triangular, with three cells and three 

 valves. Seeds woolly. 



1. T. •vUlofa. Villous Trigonia. Aubl. 388. t. 149. 



Willd. n. I Leaves obovate, downy and hoary beneath. 



.^Native of Guiana, growing chiefly by way fides, in cul- 

 tivated grounds, or in the borders of groves and thickets, 

 bearing flowers and iruit at various feafons. The Jletn is 

 fhrubby, with twining branches, fupporting themfelves upon 

 neighbouring trees, round, downy and leafy. Leaves oppo- 

 fite, ftalked, entire, three inches long and one and a half 

 broad, veiny. Stipuhis ovate, in pairs, deciduous. Clujlers 

 compound, terminal. Flowers aggregate, fmall, yellow, 

 with a red keel. Fruit three inches, or more, in length, 

 clothed with reddifh down, fplitting from the bafe. 



2. T. lavls. Smooth Trigonia. Aubl. 390. t. 150. 

 Willd. n. 2. — Leaves elliptical, fmooth and fhining on both 

 fides. — Found on the banks of a rivulet, near the bafe of the 

 hill of Courou, in Guiana. The branches of this fpecies 

 are fmooth, but twine round any thing in their way, like the 

 preceding. Leaves only one and a half or two inches long, 

 oval, entire, on (hortifli ftalks. Clujlers fevcral at the ends 

 of the branches, accompanied by fome leaves. Flowers 

 oppofite, accompanied by fmall brafteas, white, with yellow 

 anthers. Fruit about an inch long, greenifh, rather rough 

 to the touch. Seeds enveloped in foft whi^e wool. Nothing 

 is mentioned of any ufeful properties in either of thefe 

 plants. 



TRIGONIS, fo called by Jacquin, becaufe each of its 

 petals forms an inverted ifofccles triangle. See Cupania. 

 TRIGONOMETER, AuMiLLAiiT. See Armillaby 



Trigonometer. 



TRIGONOMETRY, from -rpiyo.vo;, triangle, and /tsTpov, 

 mea/ure, fignifies literally the meafure of triangles ; but it is 

 ufed here to denote that fcience which relates to the deter- 

 mination of the fides and angles of triangles, from certain 

 parts which are given. When it is applied to the folution 

 of plane triangles, it is called plane trigonometry ; and its ap- 

 plication to fpherical triangles, is called fpherical trigono- 

 metry. 



Trigonometry, from its numerous and important ufes, 

 may be confidered as one of the moft interefting branches of 

 the pure mathematics : practical and phyfical aftronomy, 

 navigation, furveying, geodefia, mechanics, in fhort nearly 

 every branch of the pure and mixed mathematics, with the 

 exception of geometry and arithmetic, are either wholly or 

 in part connefted with the principles of trigonometry ; and 

 we accordingly find that the improvements in this depart- 

 ment have kept pace with, or rather perhaps have preceded, 

 thofe which modern authors have introduced into all the 

 other branches of the exaft fciences : in faft, the trigono- 

 metry of the Greeks, and that of the moderns, which im- 

 mediately followed the invention of logarithms, and, laftly, 

 the analytical form given to it by Euler, Lagrange, &c. 

 exhibit the fame fcience under three very diftinft charafters, 



T R I 



of which it will be proper to give a flight flcetch as an intro- 

 duftion to the prefent article. 



It is very uncertain at what time trigonompf ry firft began 

 to be cultivated as a fcience, no records havmg yet been 

 difcovered which enable us to trace it to a higher age than 

 to that of Hipparchus, who flouriflied about 150 years be- 

 fore Chrift, and who, as we are informed by Theon, wrote 

 a work, in twelve books, on the chords of circular arcs, 

 which, from the nature of the title, muil have been a trea- 

 tife on trigonometry : but the earlieft work extant on thi? 

 fubjeft is the Spherics of Theodofius, in which the feveral 

 propofitions are demonftrated after the manner of Euclid ; 

 and the next in order to this, is a work by Menelaus, who 

 flourithed about the middle of the firfl; century of the Chrif- 

 tian era, and who is faid to have written nine books on this 

 fubjeft ; but of which, only three have been tranfmitted 

 down to our times. The fix that are lofl: confift^ed princi- 

 pally of tables and the nature of their conftruftion, which 

 if we poflefl'ed them would, in all probabihty, be rather 

 matters of curiofity than of real utility. The earliefl: tables 

 of trigonometry, of any importance, that we poflefs of the 

 ancients, are thofe given by Ptolemy in his Almageft, in 

 which he adopts the fexagefimal divifion of the radius, and 

 of the arc whofe chord is equal to radius, and then eili- 

 mates all the other arcs by fioths of that arc, and all the 

 other chords by 6oths of that chord. From the time of 

 Ptolemy, viz. from about the beginning of the fecond cen- 

 tury, nothing of importance, except what we owe to Theon, 

 was added to the fcience of trigonometry, till about the 

 clofe of the eighth century after Ch:ill, when the ancient 

 method of computing by chords was changed for that of 

 fines, an alteration firft introduced by the Arabians, to 

 whom we are alfo indebted for fevcral axioms and theorems 

 which are at prefent confidered as the foundation of modern 

 trigonometry ; but they ttill continued the fexagefimal di- 

 vifion ; and in this ftate it remained till Purbach, about the 

 middle of the 15th century, conltrutled a table of fines to 

 the divifion of the radius into 600,000 equal parts, and 

 computed them for every ten minutes of the quadrant ; and 

 afterwards Regiomontanus, the difciple and friend of Pur- 

 bach, carried the computation to every minute, dividing 

 the radius into 1,000,000. He alfo enriched this fcience 

 with many new theorems and precepts, which, except for 

 the ufe of logaritlims, render the trigonometry of this au- 

 thor little inferior to that of our times. 



Soon after the period here mentioned, feveral other ma- 

 thematicians alfo contributed to the advancement of this 

 fcience, either by fome ufeful alterations in the form of the 

 tables, or by other improvements ; amongil whom we may 

 mention, as the moll diftinguiflied, Werner, Copernicus, 

 Rheinold and Maurolycus : but the moft complete work 

 which had yet appeared, was publifticd by Vieta in 1579; 

 and fome other trafts on the fame fubjedl and due to the 

 fame author were publiflied by Scliooten in 1646. 



The firft part of the work to which we have above alluded, 

 was entitled " Canon Mathematicus feu ad Triangula, cum 

 Appendicibus," in which there is given a table of fines, tan- 

 gents, and fecants for every minute of the quadrant to 

 radius 100,000, with their differences ; and towards the end 

 of the quadrant, the tangents and fecants are extended to 

 eight or nine places of figures. They are alfo arranged 

 after the manner of our modern tables, increaiing from the 

 left-hand fide to 45°, and then returning backwards from 

 the right-hand to 90° ; fo that each number and its comple- 

 ment ftand upon the fame line. 



The fecond part, entitled " Univcrfalium infpeftionum 

 ad canonem mathematicum," contains, befides a regular ac- 

 count 



