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wherewith the whole Motion it has after defcribing eight thirds of its Diamoter might be generated or taken away, as the Den- sity of' the Medium to the Denfity of the Globe.
The Rejiflence of a Cylinder moving in the Direction of its Axis is not alter'd by any Augmentation or Diminution of its Length : And therefore is the fame with that of a Circle of the fame Diameter moving with the fame Velocity in a right Line perpen- dicular to its Plane.
The Refifieme of a Cylinder moving in an infinite une- laftic Fluid, ariling from the Magnitude of a tranfverfe Section; is to the Force wherewith its whole Motion while it defcribes four times its Length may be taken away or generated; as the Denfity of the Medium to that of the Cylinder, very nearly.
Hence, the Reffiences of Cylinders moving Length-wife, in in- finitely continued Mediums, are in a Ratio compounded of the duplicate Ratio of their Diameters, and the duplicate Ratio of their Velocities, and the Ratio of the Denfity of the Me- diums.
The Reffience of a Globe in an infinite unelaftic Medium is to the Force whereby its whole Motion while it defcribes eight thirds of its Diameter, might be either generated or taken away; as the Denfity of the Fluid to the Denfity of the Globe, $uam proxime.
Mr. James Bernoulli demonftrates the following Theorems.
Reffic?ice of a Triangle.— U an Ilbfceles Triangle be moved in a Fluid according to the Direction of a Line perpendicular to its Bafe ; firft, with the Vertex foremoft, and then with its Bafe ; the Iiefiftences will be as the Legs, and as the Square of the Bafe, and as the Sum of the Legs.
The Reffience of a Square moved according to the Directi- on of its Side, and of its Diagonal, is as the Diagonal to the Side.
The Reffience of a Circular Segment* Iefs than a Semicircle carried in a Direction perpendicular to its Bafis, when it goes with the Bafe foremoft, and when with its Vertex foremoft, (the fame Direction and Celerity continuing;) is as the Square of the Diameter, to the fame, lefs \ or the Square of the Bale of the Segment. Hence the Reffiences of a Semicircle when its Bafe and when its Vertex go foremoft, are to one another in a fefquialterate Ratio.
Reffience of a Parabola.— A Parabola moving in the Directi- on of its Axis firft with its Bafis, and then its Vertex foremoft, has its Reffiences as the Tangent to an Arch of a Circle, whofe Diameter is equal to the Parameter? and the Tangent equal to half the Bafis of the Parabola.
The Reffience if Vertex goes foremoft, may be thus computed : Say, as the Sum (or Difference) of the Tranf- verfe Axis, and Latus ReBum, is to the tranfverfe Axis; fo is the Square of the Latus Reclum to the Square of the Diameter of a certain Circle? in which Circle apply a Tangent equal to half the Bafis of the Hyperbola or EUipfis. — Then fay again, as the Sum and Difference of the Axis and Parameter, is to the Para- meter; fo is the arorefaid Tangent to another right Line. And farther as the Sum (or Difference) of the Axis and Parameter, is to the Axis : So is the circular Arch correfponding to the a- forefaid Tangent, to another Arch. This done, the Reffiences will be as the Tangent to the Sum (or Difference) of the right Line thus found, and that Arch laft mentioned.
In the general, the Reffiences of any Figure whatever going now with its Bafe foremoft, and then with its Vertex, are as the Figures of the Bafe to the Sum of all the Cubes of the E- lements of the Bafe divided by the Squares of the Element of the Curve Line.
All which Rules may be of ufe in the Construction of Ships, and in perfecting the Art of Navigation univerfally: As alfo for determining the Figures of the Bobs of Pendulums for Clocks, &c. See Ship, Navigation, Pendulum, <&c.
RESOLUTION, Resolutio, Solutio, in Phyficks, the Reduction of a Body into its original or natural State, by a DiA folution or Separation of itsaggregated Parts. See Dissolution.
Thus, Snow and Ice are faid to be refolved into Water ; a com- pound is refolved into its Ingredients, &c. See Snow, Com- pound, &■€.
Water refohes into Vapour by Heat ; and Vapour is again refohed into Water by Cold. See Vapour, Heat, &c.
Some of the modern Philofophers, particularly, Mr. Boyle, M. Mariotte-, Boerhaave, &c. maintain that the natural State of Water is to be congeafd, or be in Ice; inafmuch as a certain Degree of Heat, which is a foreign and violent Agent, is required to make it Fluid ; fo that near the Pole, where this foreign Force is wanting, it conftantly retains its fix'd or Icy-State. See Wa- ter.
On this Principle) the Refolution of Ice into Water* muft be an improper Phrafe. See Freezing.
Resolution, in Chymiftry, is the Reduction of a Mafs or mix'd Body into its component Parts, or firft Principles; by a proper Anaiyfis. See Principle, Analysis, &c.
The Refolution or Bodies is performed varioufly; by Difiillati- tw, Sublimation, Dijfolution-, Fermentation, &c. See each Opera- tion under its proper Article, Distillation, &c.
Resolution, in Logic, is a Branch of Method, call'd alfo Analyfis. See Method and Analysis.
The Bufinefs of the Refolution is to investigate or examine the Truth or Fallhood of a Propofition, by afcending from form; particular known Truth, as a Principle, by a Chain of Confe- quences, to another more general one, in Queftion. See Pro- position, Truth, &c,
Refolution, or the analytic Method, ftands in direct oppofition to Compoftionj or the fynthetic Method^ in which laft we defcend from fome general known Truths, to a particular one in Quefti- on. See Composition.
For an Inftance of the Method of Refolution. — Suppofe the Queftion this : Whether on the Suppofition of Man's Exiftence, we can prove that God exifts ?
To refohe this, our Method is thus. — " Mankind did not al- tC ways exift : 'Tis evident from a thoufand Confideraticns, the " Species had a Beginning; and that, according to all Hiftory, not ,c 6000 Years ago : But if it had a Beginning, there muft be " fome Caufe of its Beginning; fomething to induce it to exift
- c then more than it did before; in effect there muft bcaCaufe
a or Author of its Exiftence, for from nothing, nothing arifes :
- This Caufe, whatever it is, muft at leaft have all the Faculties
" we find in ourfelves ; for none can give more than he has ; ,c Nay, he muft ha/e others which we have not, fince he cou'd. " do what we cannot do, i. e. Create, make Man exift, dw.— " Now> this Caufe either exifts fti!l, or has ccas'd to do fo ; If tC the former, he did not exut from Eternity; for what is from " Eternity is neceffary, and can neither by ic fclf nor any other " Caufe be reduced to nothing: If the latter, it muft have been <l produced from fome other ; and then the lame Queftion will " return upon the Producer.— 1 here is then fome firft Caufe; " and this Caufe has all the Properties and Faculties we have ; ' c nay more, exifted from Eternity, &c. Therefore, from the " Suppofition of Man's Exiftence, it follows there is a God, te <&c.
Resolution, or Solution, in Mathematicks, is an orderly Enumeration of the feveral things to be done, to obtain what is required by a Problem. See Problem.
Wolf us makes a Problem to confift of three Parts. — The Pro- pofition, (which is what we properly call the Problem) the Refolu- tiottj and the Demon fir at io?i. See Proposition, &c.
The general Tenor of all Probiems is, Thofe things being done which are enjoyn'd by the Refolution ; the thing is done which was to be done.
As loon as a Problem is demonftrated, it is converted into a Theorem ; whereof the Refolution is the Hypothctis ; and the Propofition the Thefts. See Theorem, Thesis, &c.
For an Idea of the Procefs of a Mathematical Refolution } fee Resolution Algebraical.
Resolution, in Algebra, or Algebraical, is of two Kinds ; the one practiced in numerical Problems, the other in Geome- trical ones. See Algebra.
To refohe a given ?iumerical Problem algebraically, the Method is thus; i°. Diftinguifh the given Quantises from the Quantities fought; and Note the former with the firft Letters of the Alpha- bet, and the latter with the laft. See Quantity.
2°. Find as many Equations as there are unknown Quantities; if that can't be, the Problem is indeterminate; ar.d one or more of the fought Quantities may be allumcd at Pleafure— The E- quations, unlels they be concaind in the Problem it felf, are found by Theorems relating to the Equality of Quantities. See Equa- tion and Equality.
3 . Since in an Equation the unknown Quantities are mix'd with the known; it muft be reduced in fuch manner as that on- ly one unknown Quantity be found on one Side., and none but known Quantities on the other. — This Reduction is perform'd by adding the fubtracted Quantises, dividing the multiplied Quanti- ties, and multiplying the diviaed ones, extracting the Roots out of Powers, railing Roots to their Powers, <&c, fo as that the Equality may be itill prefeived. See Reduction.
To rcfihe a Geometrical Problem Algebraically.
The Procefs jih the former Article is to be obferved through- out : But as it rarely happens we come at an Equation in geo- metrical Problems by the fame means as in numerical ones; there are fome furdier things to be noted: Firft then, Suppofe the thing done which was propofed to be done.— z Q . Examine the Relations of all the Lines in the Diagram, without any re- gard to known or unknown; in order to find which depend on which; and from which being had, what others are had, whether by fimilar Triangles, or Rectangles, &c.~i°- To obtain the fi- milar Triangles or Rectangles, the Lines are to be frequently produced, 'rill they become either directly or indirectly equal to given ones, or interfect others, <&c. Parallels and Perpendicu- lars to be frequently drawn: Points to be frequently connected j and Angles to be made equal to others.
If thus you don't arrive at a neat Equation; examine the Re- lations of the Lines in another Manner.-— Sometimes 'tis not e- nough to feek the thing directly, but another thing muft be fought, whence the firft may be found.
The Equation being reduced, the Geometrical Conftructioa to be deduced there-frora 3 which is done in various Manners, in
the
