ced for the Proof of any one Problem or Theorem, the former don't always depend of the latter, get it don't readily enough appear either from the order of each, or by any other manner, when they agree together, and when not; wherefore for want of the Conjunctions and Adjectives, ergo, rurfus, c. many difficulties and occafions of doubt do often arise in reading, efpecially to thofe that are Novices. Befides it frequently hap pens, that the faid Method cannot avoid fuperfluous. Repetitions, by which the Demonftrations are oftentimes render'd tedious, and fometimes alfo more intricate; which Faults my Method doth easily remedy by the arbitrary mixture of both Words and Signs. There fore let what has been faid, touching the Intention and Method of this little Work, fuffice. As to the reft, whoever cavets to please himself with what may be faid, either in Praife of the Mathematicks in general, or of Geometry in particular, or touching the HiStory of thefe Sciences, and confequently of Euclide himself, (who digefted thofe Elements) and others

wreed of that kind, may confult other Interpreters: Neither will I (as if I were afraid leaft: thefe my Endeavors may fall short of being fatisfactory to all Perfons) alledge as an Excufe (tho I may very lawfully do it) the want of due time which ought to be employ'd in this Work, nor the Interruption occafion'd by other Affairs, nor yet the want of requifite belp for thefe Studies nor feveral other things of the like nature. But what I have here employ'd my Labour and Study in for the Use of the ingenuous Reader, I wholly fubmit to his Cenfure and Judgment, to approve if ufeful, or reject if otherwife.

J. B.

Ad amiciffimum Virum, d. C. de EUCLIDE

F

contrado, Εὐφημισμός.

A&um 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; & gloffa arctior
Stringit Theoremata; minoris anguli
Lateribus ecce totus Euclides jater,
Inclufus olim velut Homerus in nuce;
Pluteoque Jarcina 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 dumi decrefcit, ufu fit minor
Quin auctior jam evadit, & cumulatius
Contracta prodeft erudita pagina,
Sic ubere magis liquor è preffo affluit
Sic pleniori pafa inundat fanguinis
Torrente cordis Syftole, fic fuftus
Procurrit aquor ex Abyla anguftiis.
Tantilli operis ars tanta referenda unire eft
BAROVIANO nomini, ac folertia. !
Sublimis euge mentis ingentium potens!
Cui invitum nil, arduum effe nil folet;
Sic ufque pergas profpero conamine,
Radiufque multum debeat ac abacus tibi ;
Sic crefcat indies feracior feges,
Simili colonum germine affiduo beans.
Specimen futura meffis hic fiet labor.
Magnaque fama illuftria hac prahudia.

camne,

Juvenis dedir qui tauta, quid dabit fenex?

2.0

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

The

The Explication of the Signs or Characters.

Signifies

fEqual,

Greater.

Leffer.

More, or to be added.

Lefs, or to be fubtracted.

The Differences, or Excefs; Alfo, that all the quantities which follow, are to be fubtracted, the Signs not being changed.

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

The fame is denoted by the Conjunction of letters; as AB=Ax B.

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

Q &q

A Square.

Q. Q.

The ratio of a fquare number to a fquare L number.

Other Abbreviations of words, where-ever they accur, the Reader will without trouble understand of bimself; faving fome few, which, being of less general use, we refer to be explained in their places.

.The

THE FIRST BOOK

OF

EUCLIDE'S

I.

ELEMENTS.

A

Definitions.

Point is that which has 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 lies equally betwixt its Points.

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

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

VII. A plain Superficies is that which lies equally betwixt its lines.

VIII. A plain 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 contain the Angle be right lines, it is called a right-lined Angle. X. When

A

X. When a right line CG ftanding upon a right line AB, makes the angles on either fide thereof, CGA, CGB, equal one B to the other, then both thofe equal angles are right angles; and the right

line CG, which ftandeth on the other, is termed a Perpendicular to that (AB) whereon it ftandeth.

Note, When feveral 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 CG, AG make at G, is called CGA, or AĞC.

XI. An obtuse angle is that which is greater than a right angle; as ACD. XII. An acute angle is that which is lefs than a right angle; as ACB. D XIII. A Limit, or Term, is the end of any thing. XIV. A Figure is that which is contained under one or more terms.

B

XV. A Circle is a plain figure contained 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 equal the one to the other.

В

E

D

XVI. And that point is called the Center of the Circle.

XVII. A Diameter of a circle is a right line drawn through the center thereof, and ending at the cir cumference on either fide,