L 1 N

L 1 N

l ° n a Point; as, if the Point A. be mo-

vr deserif r< ^ S ? r w '^ by us Motion trace outeft Wav a L ‘ ne ; which, if it go the near-ly wecn A and 8, will be a Rjgbt or

the

nearcti W ^°^ c definition therefore will befoiiit s - g Cr foortett Difiance between any two*1 any- the Point go any way about, as

e >ther a ^ , Lines AC B, then it will trace outh*oot T[j r 00 ^Line, as the upperXC 8; or elseFro^ wp $ trn * t ones, as in the lower A C 8k‘ v eral p ^ benefis or Production of a Line’which° n ^ ec E ucncc s will fairly follow ; someI. 'p ai ? seedless Propositions in Euclide.p*t if ^ 0 wight lines cannot include a Space ;"vint, Vv ',| Vn from the fame Point to the fameL^Ways be coincident-, and drawn? ic , bo t ^ c > can only meet and make an An-4jf <w*j,. n never bound or terminate a Space.^■'Luclidc.

k:

^ 4 c a yTriangle, as AC B, any two sides,ft T\\ r( j ^ 8 taken together, are longer than3 tce 1^8; because AB is the shortest Di-1 Cetl the two Points A and 8. zo t.

3 - A >

p ^tetic an ^ cnt ( or Line touching the Cir-^t. an ^ e of a Circle, can touch it but in one

«Ci ’ ”

^ th^fv consequently will be all of it with-

“ttcU

1 6 e. 3 Euclide.

Line drawn between

V" 1 XX10.VV1A uxivvvvii "... j tWO

a ■ l nc Circumference of a Circle, falls

any

, L A11 ^ -

>thi nv , —

tv'I^’E f Circle, z e. 3 Euclid.

& ^asures, so Mr. Oughtred calls the

‘w ,'f tP? r , e Primitive Circle in the Projc-

whole Line, and both its Parts arc equal to theSquare of the whole Line. See Fig. 1.

That is, ^ a-e~% %.

or % a—a a\a e.and % e— a e~\~e e.

Therefore % %~a a-\~% a e-)~e e Q. E. D.

3. Let the Line Z^ be Cut into a-\-e j thensliall the Rectangle under the whole Line ()and the Part ( a ) be equal to the Square ofthat Part a, together with the Rectangle madeby the two Parts a and e.

That is, Z^a—an-\-ae.

-A

For X~ti-\- e

And a~\~ey.a—a a-\~ae. Q. E. D,

4. The Square of any Line, as ^, dividedinto any two Parts a and e, is equal to both theSquares of those Parts together, with the Re-ctangles made out of thole Parts.

That is, Z^ fizz a a-fiz a r-f-e e.

-2-1-s-

Multiply a-fie by its self, and the thing isplain.

4-1

narfiee-J —a e-fie e

aa-fizae-fice

sphere in Piano ; or that Line inf f< s f a )L diameter of any Circle to be Pro

^Ntor °L* Xut »hcrs, is a Line so called by itsa ft^ b„,,, T 'fi*unter, and therefore frequentlyPa er 0^*7 df»e. This is usually placed onthe back of a Sector; and running"t. .. it, you have the Artificial Lines,

COROLLARIES.

Hence ’tis plain, that the Square of any Lineequal to four times the Square of its halfFor suppose Z to be dissected, then each Partwill be a ; and multiplying a-\~a by it self, thetiling will plainly appear.

IS

m

call them

, - an d their Properties.

les. Book of Euclide treats mostly of

Li

'led

and

and

a-fia

a-fia

^f the Effects of their being divi-u7c u ° thi> c Multiplied into one another; asb r :s\. /cst Six Propositions of Book the, y < len 2ri c 7 mer of which you have here very

Hirk If t L C m st , raced Algebraically. j

P ar ctl > as > . c two Lines ^ and x; one ofas inhL 15 divided into any Number ofS * the tw n e "d" * °- The Rectangle

ftlirj all 5 . Vv “°le Lines r x, is equal to the^foL^b^anglcs made by * mulri-ne Parts of jj,

ZtdL

Thisisso

Æ , ° -j-V ine » as be divided into twole Rectangles made by the

a a~\~a a-\~a a-\—a a —4 a a,

5. If a Line be divided into two Parts equal-ly, and into two other Parts unequally, theRectangle under the unequal Parts, togetherwith the Square of (the intermediate Part) theDifference between the equal and unequal Parts,is equal to the Square of half that Line.

,.la;

Let the whole Line be 2 a, then each Part will be tu5 B 2 Let