4FF 44S F 4 1 SS-^ FF 4 ^ ^ SS 4 4 SB 
which is the i o propofitfoii of the fame book^ 
Almoft the whole dodrine of proportionals, viz. per- 
mutation, inverfion, converfion, compbficion, divifion of 
F^atio s, and proportion ex ^^uo, and confeqiiently the 
mcft ufeful propofirionsof the fth book are clearly de- 
monftrated by one definiton, and that is offimilaror 
like parts, which are (aid to be fuch as are after the fame 
manner or equally contained in their wholes ; thus (ia 
Fig. 6) the Antecedents A and C arc either equal to their 
confequents or greater, or Ids, if equal, the thing is ma- 
nifeft, if lefs, then (by the definition of proportionals) A 
and Care like parts of B and E, therefore what propor- 
tions the whole B and E have to one another, the fame 
will A and C have, which is permutation, likewife E: C : : 
B ;A which is inverlion j> lo alfo if from proportionals 
you take like parts, the remainders will be proportional, 
whence converfion and divifion are demonftrated 5 and 
if to proportionals you add like parts, the wholes will ftill 
he proportional, which is Compofition (Sec Ir the antecs'^ 
dents be greater then th^ Confequents, the Confequents will 
be like parts of them, and the demonftration exadly the 
fame with the former. 
The firft of the (5^Z> book is proved by confidering the 
generations of Parallelograms, which are produced by 
drawing or mnXu^hjin^^thQ perpendicular upon th^hajis, 
that is, takeingit fo often as there can parts and divifi- 
ons in the bafe : therefore (in Fig. 7) the fame proporti- 
on that RX fingle, hath to NX fingle, the fame hath RX 
multiplyed by XZ, that is, repeated a certain number of 
times, to NX multiplied by X Z, that is, repeated the 
fame number of times ; which is as much as to fay RX : 
NX : : par : RZ : par : NZ ; now that this proportion alfo 
is true in oblique angled parallelograms, is proved, be- 
caufe they are equal to redangled ones upon the fame 
^i?yf> and between the fame Parallels, as does thus inde- 
pendently appear (in Fig. 7) the triangles RQX and MPZ 
are eqyal, for RX=MZ, QX=PZ, RM==QP,thei^fore 
addiilg 
