do's not alwaies depend on the Former, yet when they do cohere one with another, and when not, cannot readily enough be known, either from their order or any other way; whence it not feldome comes to paffe, that through the want of Conjunctions and Adjectives, Ergo,rurfus,&c.there arifes difficulty and occafion of doubting, efpecially to fuch as are but little vers'd therein. And in the next place, it oftentimes falls out that the faid Method cannot avoid fuperfluous repetitions,whereby the Demonftrations become fometimes prolixe, and fometimes perplex'd and intricate. All which Inconveniences are easily reme-. died in our Way by the intermingling of Words and Signes at difcretion. And thus much may fuf fice to be premised concerning the Intent and Method of this Compendium. Ifhall not alledge in favour of my self the fcantneffe of time allotted to this Work, nor the avocations of affairs, nor the fcarcity of Helps to this fort of Studies amongst us (as I might not untruly) out of fear leftmy Performance fhould not throughly please every body: But I wholly fubmit to the faire Cenfure and Judgement of the Ingenuous Reader, what I have undertaken for the advantage of his Studies; to be approved, if he find it ferviceable thereunto; or, if otherwife, rejected.

Ad

Ad amiciffimum Virum,I.C.de EVCLIDE

contrado, Εύφημισμός.

FA&tum bene! didicit Laconice loqui
Senex profundus, aphorifmos induit.
Immenfa dudum margo commentarii
Diagramma circuit minutum; utque Infula
Problema breve natabat in vafto mari.
Sed unda jam detumuit; & glossa artior
Stringit Theoremata; minoris anguli
Lateribus ecce totus Euclides jacet,
Inclufus olim velut Homerus in nuce z
Pluteoque farcina modo qui incubuit, levis
En fit manipulus. Pelle in exigua latet
Ingens Mathefis, matris utero Hercules,
In glande quercus, vel Ithaca Eurus in pila.
Nec mole dum decrefcit, ufu fit minor ;
Quin auctior jam evadit, & cumulatius
Contracta prodeft erudita pagina.
Sic ubere magis liquor è presso effluit ;
Sic pleniori vafa inundat fanguinis
Torrente cordis Systole ; fic fufius
Procurrit æquor ex Abyla anguftiis.
Tantilli operis ars tanta referenda unice eft
BAROVIANO nomini, ac folertia,
Sublimis euge mentis ingenium potens!
Cui invium nil, arduumesse nil folet ;
Sic ufque pergas profpero con mine,
Radiufque multum debeat ac abacus tibi ;
Sic crefcat indies feracior feges,
Simili colonum germine afsiduo beans.
Specimen future messis hic fiet labor,
Magneque fame illuftria hæc præludia.
Iuvenis dedit qui tanta, quid dabit fenex ?

Car. Robotham, CANTAB.
Coll. Trin. Sen. Soc.

Characters.

Equall.

Greater.

Leffer.

More, or to be added.

Leffe, or to be fubtracted.

The Difference, or Exceffe; Alfo, that all the
quantities which follow,are to be fubtracted,
the Signes not being changed.

Multiplication, or the Drawing one fide of a
Rectangle into another.

The fame is denoted by the Conjunction of
letters; as ABA x B.

The Side or Root of a Square, or Cube, &c.

A Square.

A Cube.

The ratio of a fquare number to a square

number.

Other Abbreviations of words, where ever they occurr, the Reader will without trouble understand of himself; faving some few, which, being of lesse generall ufe, we referr to be explained in their own places.

THE

THE FIRST BOOK

OF

EUCLIDES

I.

ELEMENTS.

Definitions.

Point is that which hath no part. II. A Line is a longitude without latitude.

III. The ends, or limits, of a line are points.

IV. A right line is that which lyes equally betwixt it's points.

V. A Superficies is that which hath only longitude and latitude.

VI. The extremes, or limits, of a fuperficies are lines.

VII. A plaine fuperficies is that which lyes equally betwixt it's lines.

VIII. A plaine Angle is the inclination of two lines the one to the other, the one touching the other in the fame plain, yet not lying in the fame ftrait line.

IX. And if the lines which contein the angle be right lines, it is called a right-lined angle.

A:

X. When

G

X. When a right line CG ftanding upon a right line A B,makes the angles on

either fide

thereof, C GA, CGB, equall one to the other, B then both thofe equall

angles are right angles; and the right line C G,which itandeth on the other, is termed a Perpendicular to that (AB) whereon it standeth.

Note. When feverall angles meet at the fame point (as at G) each particular angle is defcribed by three letters; whereof the middle letter fheweth the angular point, and the two other letters the lines that make that angle: As the angle which the right lines C G, A G make at G, is called CGA, or AGC.

P

C

AXI. An obtufe angle is that which is greater then a right angle; as A CB. XII. An acute angle is that which is leffe then a right angle; as AC D.

D

XIII.A Limit, or Term, is the end of any thing. XIV. A Figure is that which is conteined under one or more terms.

XV. A Circle is a plain figure conteined under one line, which is called a Circumference; unto which all lines drawn from one point within the figure, and falling upon the circumference thereof, are equall the one to the other.

B

E

XV I. And that point is called the Centre of the circle.

XVII. A Diameter of a circle is a right line drawn through the centre thereof, and ending at the circumference on ei

ther