IN
(388 )
IN
Exprelfion of any fuch Tangent will be . or ■£ — ,
d — x x — d
according as we take it on the one or the other fide of the Point E 5 jn one cafe therefore it is pofitive, in the o- ther negative. But as the Points B or F approach to E, the Tangents DA and H I increafe continually, and ar the Point E they become infinite. Therefore a Quantity that continually increafeth, cannot pafs from a pofitive to a negative Expreffion, without being infinite. All this is univerfally true, whether it be a finite Quantity or a Fluxion that we confider. There is no changing from po- fitive to negative, without palling thro nothing or infinite. Applying this therefore to a Fluxion, it will follow from hence, That the Fluxion of a Quantity that exprclTeth a Maximum or a Minimum, muft be equal to nothing, or to infinite. And upon this Founda- tion, we may now eafiiy proceed to the Points of Injlctlion and Re- trogrtfiion. In order to find a ge- neral Theorem to aflift us in this Matter, let us confider the Curve A FK, (Bf.a.) whofe Diameter is the right Line A B; and its Ordi- nate* V M, eft parallel to one an- other. If through the Point F be drawn the Ordinate Applicate E F, together with the Tangent FLj and from any other Point, as M on the fame fide with A F, be drawn the Ordinate Applicate M P, as likewile the Tangent MT: It is evident, (1.) In thefe Curves that have a Point of bifiettion, that the stljciJTa A P increafes continually, and that the Part A T of the Di- ameter, intercepted between A the Origin of the Axis, and Tthe Con- courfe of the Tangent and the Axis, increafeth fuch time as the Point P fall upon E, and after it again begins to diminim : From whence it is apparent, that A T muft become the Maximum A L, when the Point P falls upon the Point E required. (2,.) In thofe Curves that have a Point of Retrogrefiion, it is evident that the Part AT increafeth continually, and that the dbfei/fa increafeth fo long, till the Point T fall upon L, after which it again di- minifheth: From whence it is clear, that A P muft be- come a Maximum, when the Point T falls upon L. N w if
A E be put =.v, EF==v, then will AL be :
whofe Fluxion, which is invariable) being divided by
y z x — y x y ~
E in the Diameter A B 5 fo that the Ordinate Applicate E F ftiall cut the Semi-cycloid in F, the Point of contra- ry Flexion. Suppofe the known Quantities ADB=5a BK = £i A B = 1 c, and the unknown Quantities A E = x, ED = 2;, the Ark AD=«, EF^j then by
the Property of the Cycloid y = z +' , and therefore
r = 3+-
But by the Property of the Circle
s ~ 1/ :
confequently a =
and u (■/ x' -}- 2,")
tuting for i, and u the a c x — ax x-\-bc'>
y-
V*-
V z c x — -xx Therefore fubfti-
r refpective Values, we have and the Fluxion thereof, (fup-
b c x — a c x • — • b c c X x
y
(fuppofing x J the Fluxion of A E muft
become nothing or infinite; that is, — JU*!. = nothing or
y %
infinite : So that multiplying by j% and dividing by — y t y will be = to nothing or infinite 5 which in the Sequel will ferve for a general Form to find F the Point of In- jkBio\ or Retrogretfion. For the nature of the Curve A F K being given, the Value of y may be found in % 5 and taking the Fluxion of this Value, and fuppofing x invariable, the Value of y will be found in x% which be- ing put equal to nothing, or Infinity, ferves in either of thefe Suppofitions to find fuch a Value of A E, as that the Ordinate EF fhall interfecT: the Curve A FK in F, the Point of InfieBhn or Retrogreifion. Of this we ihall only give one Example.
Let AFK Gtife.3.) beaprotraaedSemicycloid, whofe
TO 30
Bafe is longer than the Circumference of the generating Circle ADB, whofe Center is C. It is required to find
a y/ ac x ■ — x pofing x invariable) is _
z C X XX X/i( X — — X X
that is, ik(.)=(+ -± and C E = ~. Hence
b b
it is evident, That to have F a Point of contrary Fluxion, b muft be greater than a; for if it be lefs, then C E would exceed C B. Concerning the htfieWitn of the Rays of Light, fee Light.
INFLUENCE, a Quality fuppofed to flow from the Bodies of the Stars, or the Effect of their Heat and Light, to which the Affrologers vainly attribute all the Events that happen on the Earth, The Alchymills, who make it the Philolopher's Stone, tell us, that every thing in Nature is produced by the Influence of the Stars, which, in iheir Foliage thro' the Atmofphetc, imbibe many of its moift Parts, the grotfeit whereof they depofire in the Sands and Earths where they fall 5 that thefe filtrating thro' the Pores of the Earth, defand even to the Center, whence they are driven, by the central Fire, back again to the Surface, and in their Afcent, by a natural kind of Sublimation, as they find Earths well difpofed, they form natural Bodies, as Metals, Minerals, and Vegeta- bles, iSc ThusChymillry confining in an artificial Imi- tation of thefe natural Operations, and in applying active Principles to paffive Principles, 'tis pretended it can form natural Bodies, make Gold, ti?c
INFLUENT, a Term ufed where any Liquor or Juice, by the Contrivance of Nature, and the Laws of Circulation, falls into another Current and Receptacle. 'I husw.th refpect to the common Receptacle in thelhuman Body, the Chyle is its influent Juice, and fo is the Bile to the Gall-Bladder, and Venal-Blood to the Heart in its Diallole; and the like.
IN FORMA PAUPERIS, in Law, is when any Man that hath a jufl Caufe or Suit, either in the Chancery, or any of the Courts of Common Law, will come before the Lord-Chancellor, or Mafier of the Rolls, or either of the Chief Jul'lices, or Chief Baron, and make Oath, that he is not worth five Pounds, his Debts paid; either of the faid Judges will, in his own proper Court, admit him to fue in Forma Pauperis, and he ihall have Counfel, Clerk, or Attorney affigned to do his Bufinefs, without paying any Fees.
INFORMATION, in Law; fee Incitement. Informa- tion for the King is the fame 5 that for a common Perfon we call Declaration, and is not always done directly by the King, or his Attorney, but fometimes by another, who fucs as well for the King as himfelf; on a Breach of fome Penal Law or Statute, wherein a Penalty is given to the Party that will fue for the fame
INFORMATUS NON SUM, or Nonfum Informal; a formal Anfwer made of courfe by an Attorney, who is commanded by the Court to fay any thing he thinks good in behalf of his Client; who having nothing material to fay, makes anfucr he is not inform 'd : on which, Judgment paffes for the other Party.
INFORMIS, informed, that which has not the Form or PerfeSion it fliould have. In Aftronomy we ufe the Terms SielU Informes, for fuch Stats as have not yet been reduced into any Conftellation. Of which kind there was a great Number left by the anticnt Aflronomers, tho' Beuelius, and fome others of the Moderns, have provided for the greateft part of them, by making new Conllel- lations.
INFRACTION, a Rupture or Violation of a Treaty, a Law, Ordinance, j£c. from the Prepofition in, and the Supine of framo, to break.
INFRA-LAPSARII, the Name of a Sefl, who main- tain, that God has created a certain Number of Men on- ly to be damned, without allowing them the means ne- cefTary to fave themfelves, if they would. This Doc- trine they maintain in different manners, the Seel itfelf being divided into two Branches. Some of thetrr hold,
that
