Page:Cyclopaedia, Chambers - Volume 2.djvu/889

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T R A

or the Invention of any Number of mean Proportionals, would be of a certain Degree. See Circle, Quadb.ate.ix, Quadrature, HSic.

Whereas the different Degrees of algebraical Equations ; and, therefore, the Problem, underftood in general of any Number of Parts of an Angle or mean Proportionals ^ is of an indefinite Degree, and tranfcends ali Algebraical Equations. See Equation.

Transcendental Curve, in the higher Geometry, is fuch a one as cannot be defined by any Algebraic Equation, or which, when exprefs'd by an Equation, one of the Terms thereof is a variable orflowing Quantity. SeeCuRVE.

Thefe Curves are the fame with what 1)es Carres, and, after his Example, feveral others, call Mechanical Curves ; and which they would have excluded out of Geometry ; but Sir Ifaac Newton and M. Leibnitz-, are of another Sentiment. For, in effect, in the Conftruction of Geometrical Problems, one Curve is not to be preferr'd to another, as is defined by a more fimple Equation ; but as it is more eafily defcribed than that other. See Geometrical Line.

And fome of thefe Tranfcendental or Mechanical Curves, are found of greater Ufe than all the Algebraical ones together, except the Circle. See Mechanical.

Add, That M. Leibnitz, in the ABa Ernditor. Lipf. gives us a kind of Tranfcendant Equations, whereby thefe Tranf- cendental Curves are actually defined, and which are of an indefinite Degree ; that is, are not always the fame in all the Points of the Curve.

Whereas Algebraifts, ufe to afTume fome general Letters, or Numbers, for the Quantities fought 5 in thefeTranfcendm- tal Problems, M. Leibnitz affumes general or indefinite Equa- tions for the Lines fought 5 e.gr. putting x and y for the Abcifs and Ordinate, the Equation he ufes for a Line fought, is } a-[-bx-\-,cy-\-exy~{-fxx-~gyy, &c.=z 0. By the help of which indefinite Equation, which in reality is Finite ; for it may always be determined, how far foever 'tis necefTary to raife it ; he feeks the Tangent ; and that which refults, com- paring with the given Property of Tangents, he finds the Value of the affumed Letters, a, b, c, f£c. and thus defines the Equation of the Line fought.

If that Comparifon abovementioned, do not proceed, he pronounces the Line fought, not to be an Algebraical, but a Tranfcendental one.

Thisfuppofed, he goes on to find the Species of Tranfcen- dency ; for fome Tranfcendentals depend on the general Divi- sion or Secfion of a Ratio, or upon the Logarithms; others upon the Arks of a Circle, and others on more indefinite and compound Enquiries.

Here, therefore, befides the Symbols x and 4', he affumes a Third, as v, which denotes the Tranfcendental Quantity ; and of thefe Three, forms a general Equation for the Line fought - 7 from which he finds the Tangent, according to the differential Method, which fucceeds even in Tranfcendental Quantities. What he finds he compares with the given Pro- perties of the Tangent ; and fo difcovers, not only the Values of a,b, c, &c. but alfo the particular Nature of the Transcen- dental gktantity.

And though it may fometimes happen, that the feveral Tranfcendents are fo to be made ufe of, and thefe of different Natures too, one from another 5 alfo though there be Trans- cendents of Transcendents, and a Progreffion of thefe in In- finitum, yet we may befatisfied with the moft eafy and ufeful one -j and for the moft Part may have recourfe to fome pecu- liar Artifices for fhortning the Calculus, and reducing the Problem, to as fimple Terms as may be.

This Method being applied to the Bufinefs of Quadratures, or to the Invention of Quadratrices, in which the Property of the Tangent is always given ; 'tis manifest not only how it may be discovered, whether the indefinite Quadrature may be algebraically Impoflible, but alfo how, when this Im- pofTibility is difcovered, a Tranfcendental Quadratrix may be found ; which is a Thing which had not before been fhewn. So that it feems, that Geometry is by this Method carried infinitely beyond the Bounds, to which Fietaand Des Cartes brought it: Since by this means a certain rnd general Analyfis is eftablifti'd, which extends to all the Problems which are of no certain Degree, and confequently not comprehended within Algebraical Equations,

Again, in order to manage Tranfcendental Problems, where-ever the Bufinefs of Tangents or Quadratures occurs, by a Calculus, there is hardiy uny that can be imagined fhorter, more advantageous,or univerfal, than the Differential Calculus, or Analyfis of Indivifibles and Infinites.

By this Method we may explain the Nature of Tranf- cendental Lines by an Equation, e. gr. Let A be the Ark of a Circle, and x the verfed Sine : Then will A = Jdx

=
and if the Ordinate of the Cycloid bey, then will 1/2*-- xx

■ fdx y = \/ 2X — ##-}-=—=—=: which Equation perfectly </ 2X — xx

expreffes the Relation between the Ordinate y and th£ Abcifla x, and from it all the Properties of the Cycloid may be demonftrated.

Thus is the Analytical Calculus, extended to thofe Lines which have hitherto been excluded, for no other Caufe, but that they were thought incapable of it, See Geome- trical Line.

TRANSCOLATION, in Pharmacy, the fame with Filtration or Percolation. See Filtration, ££?c.

TRANSCRIPT, a Copy of any Original Writing; par- ticularly that of an Act or Inftrument, inferred in the Body of another. See Copy.

In this Senfc, we fay Tranfcript of a Fine, £5jc. See Fine, Duplicate, l$c.

TRANSCRIPTIO Recognitions faBtS coram Jpftkiariis Itinerant ibzts, l£c. is a Writ for the certifying a Recognizance into Chancery, taken before the Juftices in Eyre. See Recognizance.

Tr a n sc r i p t 10 pedis finis levati mitte?2da in CanceUariam, is a Writ for the certifying the Foot of a Fine, levied before the Juftices in Eyre, &c. into Chancery. See Fine.

TRANSEAT, in the Schools, &c. a Term purely Latin, fignifyingZe? itf>afs, or fuppofe a Propofition to be ttue, with- out granting it.

Hence the Proverb, Tranfeat, Gr<ecmn ejf, non legitur : The Diftion is faid to have taken its Rife from fome ancient Commentators, or Gloffographers of the Civil Law, who not underftanding Greek, pafs'd over all the Words that occurr'd in that Language^without explaining them.

In the Roman Chancery, a Nil Tranfeat is a kind of Oppo- fition made to the fealing of a Bull, or ro the Delivery of fome other Inftrument, till the Parties againft whofe Interefl it is directed, have been heard againft it,

TRANSELEMENTA'TION, in the Schools, a Change of the Elements or Principles of one Body into thofe of an- other. See Element.

Such is that which the Roman Catholicks contend for m the Eucharift 5 where the Elements of Bread and Wine are changed into thofe of Flefh and Blood. See Transue- stantiation.

Tranfelementation, where-ever it happens, is always allow'd Miraculous ; or an EffecT: beyond the ordinary Powers of Nature. See Miracle.

TRANSFER, in Commerce, &c. an A& whereby a Perfon furrenders his Right, Interefl: or Property in any thing moveable or immoveable, to another.

Trie Term is principally ufed in the Commerce of Stocks ; for the afligning and making over Subfcriptions, or Shares therein, to fuch as purchafe them of the Proprietors. See Subscription, £f?c.

In the South Sea Company, the Smxk, Eaft India, ' £-fc. Transfers are made, by erafing the former Proprietors Names, and entering the Stock under the Name of the Purchafer, under his proper Letter of the Alphabet. See Company, Bank, &c

In order to a Transfer-, 'tis required the Party bring another with him to fwear he is the fame Perfon to whom the Stock is enter'd. A Counterfeit in this Cafe is by Aci: of Parlia- ment made a capital Crime.

The Lawyers fay, that the Sale, or Donation of an In- heritance, &c. Transfers the Property, Rights, &c. thereof

TRANSFIGURATION, among Divines, that miracu- lous Change wrought by Jefus Chrijt, in Prefence of St. ^Peter* St. James and St. John, on Mount Thabor, where he appear'd. in his Glory, in the middle of Mofes and Elias. See the Defcription thereof in St. Matthew, c. xvii.

The Term is alfo applied to a Feafl held in the Romip Church on the Sixth of Augnfl, in Commemoration of that Miracle.

TRANSFORMATION, a Change of Form ; or a Meta- morphofis. See Metamorphosis.

Tne Chymifls have been a long Time feeking the Tranf- formation of Metals, that is, their Tranfmutation, or the Manner of changing them into Gold. See Transmuta- tion.

Among the Myfticks, by Transformation, is underftood a Change of the Contemplative Soul, whereby it is in fome meafure deified or converted into the Subftance of God, and wherein it is, as it were, loft andfwallow'd up in the Divinity, fo as not to perceive its own Diftinction from God.

The Senfe of the Word Transformation is very liable to be abufed ; and, in effect, the Quietifts and Quakers have abufed it. But many of the Myfticks ufe it innocently enough ; meaning no other by it than what St. 'Paul did when he faid, Vivo ego, jam non ego t vivit veto hi me Chripis.

Transformation is alfo fometimes ufed for what we more properly czllTransfubfiantiation. See Transubstan- tiation.

TRANSFUSION, the Aft of pouring a Liquor out of one Veflel into another.

The