Page:Cyclopaedia, Chambers - Volume 2.djvu/103

 LIN

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LIN

LlNCTUS, a Form of Medicine the fame as Lamba- the, and probably from the fame Derivation ; or from Lingua the Tongue, as being to be licked up with the Tongue. See Eclegma.

LINE, a fmall French Meafure, confiding of the 12th part of an Inch, or 144th part of a Foot. The Geome- tricians, notwithstanding its Smallnefs, conceive it divided into fix Points-

LINE, in Genealogy, is a Series or Succeffion of Re- lations in various Degrees, all defcending from the fame common Father. Diretl Line is that which goes from Fa- ther to Son, which is the Order of Afcendants and De- fendants. Collateral Line is the Order of thofe who de- scend from fome common Father related to the former, but out of the Line of Afcendants and Defcendants. In this are placed Uncles, Aunts, Coufins, Nephews, g?c.

LINE, in Geometry, is a Quantity extended in Length only, without either Breadth or Thicknefs, and is formed by the Motion of a Point. There are two kinds of Lines, viz,. Right Lineiznd Curve Lines. Thus if the Point A move towardsB, Cr%i' Plat. Geometry) byits Motion itdefcribes a Line ; and this, if the Point go the neareft way towards B, will be a Right or Streight Line, whofe Definition therefore will be the neareft or ftiorteft Diftance between any two Points, or a Line all whofe Points tend the fame way. If the Point go any way about, as in any of the Lines A C B, A c B, it will trace our either a crooked Line, as the upper A c B i or elfe two or more ftreight ones, as in the lower ACB.

Lines considered as to their Pofitions, are either Pa- rallel, Perpendicular, or Oblique j the Conftru&ion and Properties whereof, fee under Parallel, Perpendicular, ckc.

Euclid in his fecond Book treats mofily of Lines, and of the Effecf s of their being divided, and again multiplied into one another; the Subftance of his Doctrine may be thus demonftrated Algebraically. (1.) If there be two Lines 2. and x j one of which, as as, is divided into any number of Parts, as into a -L. e ~\~ i ~\- 0, the Reef angle under the two whole Lines ax is equal to the Sum of all the Reef angles made by x multiplied into the Parts of as.

zi * I • I j I ; x| I

that is, z> x = x a -j- x e-j- xi -\~x o. This is fo plain, it needs no proof. (2.) If a Right Line, as z, be di- vided into two Farts a -f- e, the Rectangles made by the whole Line, and both its Parts, are equal to the Square of the whole Line; That is, z,a-~ze—zz. For z a^=-a a ~\- a e ; and »e = «e-f ee; therefore zz = a a -j- za -j- e e. (5.) Let the Line z be cut into a-\- e ; then fhall the Rectangle under the whole Line(Z) and the Part (a) be equal to the Square of that Part a, together with the Rectangle made by the two Parts a and e j that is, Z fl = a a -\~a e. Z ■*

[ e I For Z = a -}- e. And a ~\- ex a = a a

+ a e. (4.) The Square of any Line, as Z, divided into any two Parts, a and e, is equal to both the Squares of thofe Farts, together with the Rectangles made out of thofe Parts, that is, Zz=^aa~f-zae~\~ee.

Z. * I e. I Multiply a~\~e

by itfelf, and the Thing is plain. a 4~ e a -j- e

aa -j- e e

-|- a e -f- ee

Let the flrft Lhie be 2 a, and the Part added e, then the whole will be 2 a -j-?, which multiplied by e t produceth 2 a e -|- ee ; and the Square of half the Line a a being added to it, it will be zae -j- ee -\~ a a, which is equal to the Square of a + f - (7.) If a Quantity or Line be di- vided any how into two Parts, the Square of the Whole added to the Square of one of the Parts, mall be equal to two Rectangles contained under the whole Line, and that part, added to the Square of the other Part.

Z i__ - I I I

Let a be one Part, and e the other 3 the Square of the Whole, and of the lelTer Part e makes a a -L. zae-\- zee. then if the Whole a -j- e be multiplied by twice e, it will produce 2 a e -j- 2 e e ; and if to this be added the Square of the other Part a a, the Sum will b& aa-\- zae-\- zee, equal to the former. (8.) If a Line be cut anyhow into two Parts, the Rectangle under the whole Line and one of the Parts taken four times, and added to the Square of the other Part, is e^ual to the Square of the Whole, and the other Part added to it, as if it were but one Line.

a a -J- 2 a e-j-ee. Hence it is plain, that the Square of any Line is equal to four times the Square of its Half. For fuppofe Z to be bife&ed, then each Part will be a 5 and multiplying a --a by itfelf, the thing will plainly appear.

a -j- a

a a-\~ au~\- & a ~\-aa = 4 a a. (5.) If a Line be divided into two Parts equally, and into two other Parts unequally, the Rectangle under the un- equal Parts, together with the Square of the interme- diate Part, will be equal to the Square of half that Line. Let the whole Line be 2 a, then each Part will be a. Let the lefTer unequal Part be e, then the greater unequal Part will be za — e; which multiplied bye, produces zae — ee: To which adding the Square of the Difference or intermediate Part a — e, which is fid — 2 fle+ee, the Sum will be only a a, the Square of half the Line. (<5.) If a Line be bifected, and then ano- ther Right Line be added to it, the Rectangle or Product of the whole augmented Line multiplied by the Part added, together with the Square of the half Line, will be equal to the Square of the half Xi we, and Part added, as one Line. ,

? - I - 1 i

Let the whole Line be a~\~e, then four times that multi- plied by e (or the Quadruple Reclangle under that and e) will be 4fie-}~4 ee j to which adding the Square of the other Part a a t the Sum will befi<*-f-4 fle ~-f"4 ee * And if you fquare fl-j-2e, which exprefleth the whole Line with e added to it, the Product will be the former Sum of aa ~\- a 4«e -j-4ee. (9.) If a Line be bifefted, and alfo cut into two other unequal Parts, the Sum of the Squares of the unequal, Parts will be double the Sum of the Squares of the half Line, and of the Difference be- tween the two unequal Parts. Let the whole Line be 2 a, and the Difference between the equal and unequal Parts b 5 thea the greater unequal Part will be a-\-b, and the leffer a — b: The Sum of the Squares of the unequal Parts will be 2 aa~\> zbb, which is double to the Square of half the Line added to the Square of the Difference. (10.) If a Line be bifefled, and then another Line added to it j the Square of the- whole increafed Line y together with the Square of the Part added, is double the Sum of the Squares of the half Line, and of the half Line and Part added, taken as one Line.

1.

1.

Let the whole Line be 2 a, and the Part added e j then the whole increafed Line will be 2 a -\- e, and the half Line and Part added will be «-f-e; the Sum of the Squares of 2 a -J- e, and ofs, \si^aa-\-\ae~\'zee^ which is plainly double to a a, and a a -\- zae — ee ad- ded together.

LIlSiE, in Geography and Aftronomy, is ufed by way of Eminence for the Equator or Equinoctial Line, which, in the Heavens, is a Circle defcribed by the Sun in his Courfeon the 21ft Day of March, and the 2.1& of Septem- ber. On the Earth 'tis an imaginary Circle, anfwering to that in the Heavens. It divides the Earth from Eaji to Weft into two equal Parts, and is at an equal Diftance from the two Poles j fo that thofe who live under the Line, have the Poles always in their Horizon. The Lati- tudes commence from the Line. The Seamen ufe to duck their Paffengers the flrft time they cut the L'we.

LINE, in the Art of War, is underftood of the Difpo- lition of an Army ranged in Order of Battel. An Army ufually confifts of three Lines ; the flrft is the Front, Van, or Advance Guard ; the Main Body forms the fecond, in which is the General's Poft •■> the third is a Referved Bo- dy or Rear-Guard. 'Tis a Rule to leave 15c Paces di- ftance between thefirft Line and the fecond, and twice as much between the fecond and third, to give room for rallying.

Line is alfo underftood of the Difpofition of a Fleet on the Day of Engagement 3 on which occafion the VefTels are always drawn up in one Line. A Ship of the Line, is a Veffel large enough to be drawn up in the Line, and to have place in a Sea- Fight. /

LINE of Demarcation, or Alexandrian Line, is a Me- ridian pafling over the Mouth of the River Maragnon, and by the Capes of Hoatnas and Malabrigo. 'Tis^fo callM from Pope Alexander VI. who to end the Difputes be- tween the Crowns of Caftile and Portugal, about their Boundaries in 1493, drew an imaginary Line on the Globe, which was to terminate the Prctenflons of each. By which Partition the Baft-Indies fell to the Lot of the Fortugtteje, and iheWeft-Indies, thennewly difcovered, to the Caftilians.

LINE of Direflion, in Mechanics, is that, according to which a Body endeavours to move. The Term is alfo ufed tofignify the Liwethat pafles thro' the Center of Gra- vity of the heavy Body to the Center of the Earth ; which mull alfo pafs thro' the Fulcrum 6t Support of the heavy Body j without which it would fall.

LINE,