Lemma 
If in a right Lmc, at any diftances, be plac'd any num. 
ber of things, abed, &c. the number of the Intervals 
is.onejl^ff tfeiij the nurTi|)er of fbihgs. 
For, whereas every Interval is terminated by two ad- 
jacent things, if to any number of things, be added bne 
thing more, one Interval only is thereby added. Q. E.D. 
Lemma, 2d. 
. The number of the Alternations of m things abed, &c. 
different each from cther^ trken m and m, is iii fimC^ lae 
number of the Alternations of m—i things a be, taken 
m^-i and m— i. 
For^ f by Le^. iftj the laft Letter d, befidcs ttic^ ..»i:-oa 
it hath,., may have m— ^2 pofitions, viz, in the.Inccrvals 
which are between m—i rHihgs ab'c^ but it 'may alfo 
have one more, for it may be put firft of all, it/may there- 
fore have m pofitions ^ and thofe in all (he different Or- 
ders, whereof ui— I things are eapable 5 wbicli being all 
the pofiible pcfitions of d, -in all the Varieties of a b'c, is all 
the variety v/her6of the whole number of thin'gs expofed 
abed, d^^V is capable. Q. E. EX 
Lemma j^d. 
The number of the Altern^ions of m tbings a b e d, &c. 
different each from other, taken m and ni, is equal to m x 
m— mm— 2 ^ m--^ x m—^,6\\ contlnu'd to m-places. 
For let rriO, exprefs the number of the Alternations of 
m rhings different each from other 5 m—i O, of m— i 
things and the li^e. 
'Tis evident (hat if m ~ i. It will be m O — : ni5 for 
there can be but one order of one thing. 
And if m be greater than unity, then will it be(by Lem.2) 
uiO =m — lO =^m^ m^i « m~2 O = m x m- i « 
^ m— 2 
