Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

PROBLEM I.

HAving the sum of any two sides given for forming a Triangle ABC, to find each of the sides, and form the Triangle.

Supppose e. g. three Lines given in Fig. 14. the first =AB+AC in the Triangle sought, the second =AB+BC, the third =BC+AC, to find each of the sides e. g. to find AB, which being known, the rest will be so also.

SOLƲTION.

1. Denomination. Make AC+AB=a; AB+BC=b; BC+AC=c; AB=x; then will AC=a−x, and BC=b−x, and so the Denomination be compleat.

2. Equation. Now if the values of the two last Lines BC and AC be added into one Sum, which we had before given; you'l have this Equation a+b−2x=c.

3. Reduction. By adding on both sides 2x, you'l have 〈 math 〉〈 math 〉; and substracting from both sides 〈 math 〉〈 math 〉; and dividing both sides by 〈 math 〉〈 math 〉.

4. The Effection or Geometrical Construction, which the Equation thus reduced will help us to

Join AE=a and ED=b in one Line AD, and from this backwards cut off DF=c; and divide AF which remains into two equal parts in B, and you'l have AB the first side of the Triangle to be formed; and BE will give the other side AC, which substracted from ED, will leave GD= to the third side BC; of which you may now form the Triangle ABC.

5. A general Rule for Arithmetical Cases. Add the two former Sums, and from the Aggregate substract the third Sum; half the Remainder will give the side AB common to the two former Sums. For an Example take this Question: There are three Towns of ancient Hetruria, viz. Forum Cassii (which the Letter A denotes in ▵ ABC) Sudertum (B) and Volsinii (C) which are at this distance one from another; if you go from Volsinii to Forum Cassii and thence to Sudertum, you mst go 330 Furlongs; from Forum Cassii to Sudertum and thence to Volsinii there are 306 Furlongs; lastly, from Sudertum to Volsinii and thence to Forum Cassii 272 Furlongs. How far is each Town distant from each other.