164 Improved Geometrical Plotting Quadrant, &c. 



try, in teaching, either hy construction or calculation, the 

 knowledge of all the properties of relations between the 

 three sides and three angles, of which every plane triangle 

 is composed. Euclid having demonstrated, in the fourth 

 proposition of the sixth book of his Elements, that in any 

 two similar triangles (by which he means their having the 

 same angles, without regard to the actual lengths of their 

 sides, for one triangle mav be very small and the other ever 

 so large,) every pair of the corresponding sides in the two 

 triangles are proportional ; it is the business of trigonometry 

 to solve such problems, with the help of the tables of sines 

 and tangents, or of sectors, sliding or other rules, and scales, 

 by which you can rind, on inspection, a right-angled tri- 

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

 having one of its angles equal to 90°) which can be pro- 

 posed, 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 triangle under consi- 

 deration, so is any other side of the same tabular triangle 

 to the corresponding side supposed to be sought of the tri- 

 angle in question. It is evident, that by means of the base 

 line, perpendicular, and either the upper or lower limb of 

 mv instrument, by the two motions of which the perpen- 

 dicular is capable, and the angular motion of which the 

 iimbs are capable, any right-angled triangle whatsoever, as 

 CBE or CDE, in the diagram fig. 6. may be instantly 

 formed (by bringing the top corner of the perpendicular to 

 touch the limb) with th» same or greater facility than it 

 could be taken out of a trigonometrical table, measured by 

 the compares on the sector, or set on any instrument now 

 m use for that purpose. But no instrument that I have seen 

 or read of is capable of forming immediately any obtuse- 

 angled triangle, as on my geometrical plotting quadrant can 

 be done ; nor can the trigonometrical 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. I shall therefore 

 give ihe solution of five problems. First, supposing that 

 fig. (i. represents my instrument, set to answer this and the 

 1 following 



