g20 PLOTTING QUADRANT, LEVEL, AND CALCULATOR, 



metry, in teaching, either by construction or calculation, the 

 knowledge of all the properties or relations between the three 

 sides and three angles, of which every plane triangle is corn- 

 Similar trian- P^s^*^' Euclid having demonstrated, in the fourth propo- 

 fles propro- sition of the sixth book of his Elements, that in any two 

 tlie'r' "to * similar triangles (by which he means their having the sQm«^ 

 angles, without regard to the actual lengths of their sides, 

 for one triangle rnay be very small and the other ever so 

 large) every pair of the corresponding sides in the two trian- 

 V t* II - ) ^^^ ^^^ proportional ; it is the business of trigonometry to 

 plied to rectan- solve such problem.s, with the help of the tables of sines and 

 j^es, tangents, or of sectors, sliding or other rules, and scales, by 



which you can find, on inspection, a right-angled triangle, 

 exactly similar to any given right-angled triangle, (or having 

 one of its angles equal to 90°) which ca?i be proposed, or can 

 occur in practice ; and by the Rule of Three we say, as any 

 side of the tabular triangle is to the similar side, supposed to 

 be known, of the triani>ie under consideration, so is any 

 other side of the same tabular triangle, to the corresponding 

 side supposed to be souo-ht, of the triangle in quesi-ion. It is 

 _ t V). ^^'^^^'^^' *^^*' hy means of the base line, perpendicular, and 



pUcableto either the upper ^ or lower limb of my instrument, by the two 

 t^esej motions of which the perpendkidar is capable, and the an- 



gular motion of which the limbs are capable ; any right- 

 angled triangle whatsoever, as C B E, or C D E, in the 

 diagram Fig. 6, P/. V, may be instantly formefl, (by bring- . 

 ing the top corner of the perpendicular to touch the lirnb) 

 with the same or greater facility, than it could be taken out 

 of a trigonometrical table, measured by the compasses on the 

 sector, or set on any instrument now in use for that purpose. 

 But no instrument that I have seen or read of is capable of 

 and ennally to forming immediately ani/ obtnse-ttvgled triangle, as on my 

 cbiuse angles, geometrical plotting (jvadrant can be done; nor can the tri- 

 gonometrical tables be applied, to produce the sides and 

 angles of such a triangle, without some trouble, in any case ; 

 and in some of the most useful cases in practice, the labour 

 is very considerable. 1 shall therefore give the solution of 

 Prob1ems(6be live problems. First, supposing, that Figure 6, PLY, 

 solved by it. represents my instrument, set to answer-ihis and the follow- 

 ing problems ; A, B, C, being the triangle under consider- 

 ation ; 



