T R I 



with double of the fquare of the femi-bafe, is equal to the 

 fquares of both the fides taken together. 



19. A whole triangle is to a triangle cut off by a right 

 line, as the reftangle under the fides cut off is to the re&- 

 angle of the other two fides. 



20. In a right-angled plane triangle A D E ( Plate X. 

 Geometry, fg. 17.) a line D B drawn from the right angle 

 or the vertex, perpendicular to the hypothenufe, divid?s 

 the triangle into two other right-angled triangles, A D B, 

 D B E, which are fin-iilar to the firft triangle, and alfo to one 

 another. 



Hence it follows : 1. That EB:BD::BD:BA; 

 and A E : E D :: E D : E B ; and A E : A D :: A D : 

 A B. (See Proportional.) Whence, 2. As the angle in 

 a femicircle is a right angle, it follows that, if from any 

 point D in the periphery of a femicircle A D E, a perpen- 

 dicular D B be let fall upon the diameter A E, and from 

 the fame point D, to the extremities of that diameter, two 

 chords DA, D E, be drawn ; the fquare of thtit per- 

 pendicular will be equal to a reftangle under the two feg- 

 ments of the diameter ; and the fquare of each chord 

 equal to a reftangle under the whole diameter and its 

 adjacent fegment : i. e. B D' = E B x B A ; E D' = E 

 B X E A ; and A D= = A B X E A. 



21. In every right-angled triangle, the fquare of the 

 hypothenufe is equal to the fum of the fquares of the other 

 two fides. See Hypothenuse and Subtense. 



22. If any angle of a triangle be bifefted, the bifefting 

 line will divide the oppofite fide, in the fame proportion as 

 the legs of the angle are to one another. 



23. If the vertical angle of any triangle be bifefted, the 

 difference of the reftangles, made by the fides and the 

 fegments of the bafe, is equal to the fquare of the line that 

 bifefts the angle. Thus, if a right line B E (Plate XV. 

 Geometry, Jig. 15.) bifeft an angle ABC of a triangle, 

 the fquare of the faid line B E = A B x B C - A E 

 X E C. 



24. To divide a triangle into any given number of equal 

 parts; divide the bafe C D {Jis- 16.) into as many equal 

 parts as the figure is to be divided into ; and draw the lines 

 A I, A 2, &c. 



2J. If in fimilar triangles from any two equal angles to 

 the oppofite fides two right lines be drawn, making equal 

 angles with the homologous fides ; thofe right lines will 

 have the fame ratio as the fides on which they fall, and 

 alfo divide thofe fides proportionally. 



26. If in two triangles having one fide common to both, 

 from any point in that fide, two lines refpeftively parallel 

 to two contiguous fides be drawn, terminating in the two 

 remaining fides, thofe Unes will have the fame ratio as the 

 fides to which they are parallel. Hence, if thefe fides are 

 equal, thofe lines will be alfo equal. 



27. If through any point within a triangle three right 

 lines be drawn, from the angular points to cut the oppofite 

 fides, the fegments of any one fide will be to each other, as 

 the reftangles under the fegments of the other fides taken 

 alternately. Hence if the former fegments be equal, the 

 forementioned reftangles will be equal, and therefore the 

 fides of the triangle cut proportionally, and a hne connefting 

 the points of divifion will be parallel to the bafe. 



28. Triangles having one angle in the one equal to one 

 angle in the other, are in the ratio of the reftangles con- 

 tained under the fides, including the equal angles. Hence, 

 if the reftangles be equal, or the fides reciprocally propor- 

 tional, the triangles will be equal. 



Triai^GLES, Properties of Spherical. Sec SPHERICAL 

 Triangle. 



T R 1 



Triangles, Solution of. See Trigonometry. 



Triangle, an iron mufical inftrument with three fides, 

 which ferves as an accompaniment to other inftruments in a 

 military band, and in the ftreets : the performer fupports it 

 by a ring at the top with his left hand, and beats it with 3 

 fmall iron rod in his right hand. At the loweft angle iron 

 rings are placed, v/hich by their vibration augment the 

 found. 



Triangle IJland, in Geography, an ifland of South Ame- 

 rica, in the mouth of the Oronoko, where the French fettled 

 a faftory in the year 1765. — Alfo, one of the fmaller Ba- 

 hama iflands, fo called. N. lat. 20° 51'. W. long. 69° 53'. 



Triangles, a dangerous fhoal in the Eaft Indian fea, 

 neai- the N. coaft of the Pracel, or Prafil. 



Triangles, Southern, a reef of rocks and iflets in the 

 bay of Honduras. N. lat. 17° 45'. W. long. 88° 40'. 



TRIANGULAR Apple-Ladder, in Rural Economy, a 

 ladder of this fort for gathering apples from the trees with 

 eafe and without brnifing them. It is about eighteen feet 

 in height, and has two other branches, which are each of 

 the fame length, fattened by iron hoops or rings at its top- 

 part. Thefe parts all diverge from each other when the 

 ladder is in ufe, and appear fomewhat in the manner of the 

 corner rafters of a triangular roof, forming a fort of tri- 

 angle. At about four feet from the ground, each branch 

 and the ladder part has a hook fixed to it, for the purpofe 

 of ftretching out a triangular cloth by ; in the middle of 

 which is formed a circular funnel of the fame material. The 

 cloth has at each corner a leather ftrap, pierced with a num- 

 ber of holes, in order that an equal degree of tenfion may 

 conftantly be given, whether the ladder and its branches be 

 much extended or not. 



In coUefting the apples, the gatherer, afcending the lad- 

 der, throws the fruit as he ftrips it from the boughs of the 

 trees into the cloth, whence it rolls down the funnel part 

 into the bafket which is placed to receive it below. 



So much injury and mifchief are done to apple-trees at all 

 periods of their growth, by fetting ladders againft the boughs 

 of them, and the fruit is fo greatly bruifed and depreciated, 

 as well as fubjefted to decay, by gathering it in the ufual 

 modes, that both praftices (hould be diicontinued, and 

 better ones, fuch as the above, be had recourfe to in fuch 

 cafes. 



Triangular Battalion, in the Military jlrt. See Bat- 

 talion. 



Triangular Canon. See Canon and Sine. 



Triangular Compajfes, are fuch as have three legs or 

 feet, by which to take off any triangle at once : thele are 

 much ufed in the conftruftion of maps, globes, &c. See 

 Compasses of three legs. 



Triangular Fort. See Fort. 



Triangular Leaf, in Botany. See Leap. 



Triangular Numbers, are a kind of polygonal num- 

 bers ; being the funis of arithmetic progreflions, the dif- 

 ference of whofe terms is i. 



Thus, Of arithmetical progreff. 123456 

 are formed triang. numb. i 3 6101521 



For the rationale and management of thefe numbers, fee 

 Malcolm's Arith. book v. ch. 2. 



Triangular Quadrant, is a feftor furnifhed with a loofe 

 piece, by which to make it an equilateral triangle. 



The calendar is graduated on it, with the fun's place, 

 decUnation, and other ufeful lines ; and by the help of a 

 firing and a plumbet, and the divifions graduated on the 

 loofe piece, it may be made to ferve for a quadrant. 



Tkianculah iVindtng-Zlairs. See Stair. 



TRIAN. 



