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SYMBOLICAL NOTATION AND ABBREVIATIONS.

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I.-Signs common to Arithmetic, Algebra, and Geometry. .: because.

+ plus, add, together with. .. therefore.

minus, subtract, take away. :: wherefore.

~ difference between. equals, or equal.

x into, multiply. # not equal to, or unequal.

by, divide. greater than.

root. * not greater than.

ratio. less than.

:=: equality of ratios. * not less than.

::: : proportion. : : : Numbers or Quantities in Progression The signs >, $i <, K, between ratios, as A: B > C: D, or

A: B C: D, or A:B < C : D, or A:B XC: D, denote that the one ratio is greater than, or not greater than, less than, or not less than, the other ratio, according to the sign.

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II.-Geometrical Signs. a point.

A triangle. | straight line.

parallelogram. parallel, parallel to.

square, Drectangle. L angle.

O circle.
I perpendicular to, at rt. Zs. Oce circumference.

* When an s is added to a sign, or to an abbreviation, the plural is denoted A single capital letter, as A, or B, denotes the point A, or the

point B; but sometimes, as in Bks. V and VI, the quantity, or

magnitude, A, B, C, &c. Two capital letters, as A B, or CD, denote the straight line A B, or

CD; but when the letters indicate opposite angles, they denote a parallelogram, or a rectangle, or a polygon, as the

figure will show. A capital letter, or two capital letters, with the numeral 2 just above

to the right hand, as A2, or A B”, denote not the square of A, A B, but the square on A or A B.

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Capital letters, with a point between them, as AB.CD, denote,

not the product of AB multiplied by CD, but the rectangle formed by two of its sides meeting in a common point.

III. - Additional Algebraic Expressions.

n or p

M

Magnitude. m multiple.

mtn m A &c.

multiple of A &c. m A, m B, &c. equimultiples of A,B mn

&c. m (A + B) multiple of (A+B) (m + n) A m (A-B) multiple of (A-B). m (A+B-C) multiple of the excess pt.

of (A+B) above C. 'sub-m

mn A

another multiple. the sum of the quanti

ties m & n. the product of m X n. a multiple of A by mn a multiple of A by

m + n. part. submultiple.

III.-Abbreviations.

.......Product.

Add ... ...Addendo, by adding.
App..... Application of a Prop.
Appl......... Applicando, by applying
C. or Con..Construction.
C. 1 &c. .. Step 1 &c. of the Con-

struction.
Conc.. Conclusion, inference.
Cor. . .Corollary.
Dat... ....Datum, or data.
D. or Dem. Demonstration.
D. 1 &c. ..Step 1 &c. of the Dem.
E. or Exp..Exposition, or Particular

Ênunciation of a Prop. Ex.

Example.
Gen. .. General Enunciation.
H. or Hyp.Hypothesis of a Prop.
H. 1 &c... Step 1 &c. of the Hyp.
L

..Line.
M. or Mag. Magnitude.
P. or Prop. Proposition.

Pon......... Ponendo, by placing, by

position. Prob.... ..Problem. Proced ...Precedendo, by going on. Prel. . Preliminary Prod.. Pst. or Psts.Postulate, or Postulates Quæs. .. .. Quæsitum or Quæsita. R... .Ratio. Rec.......... Recapitulation.

.Remark to be made. Sch.. .. Scholium or Scholia. Sim.. .So, similarly, by similar

reasoning S. or Sol... Solution of a Problem. Sum. . Sume, take away. Sup.. .Suppose, or Let. Super..... Superponendo, by super

dosition. Theor..... Theorem.

Remk. . .

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Q. E. D. quod erat demonstrandum, which was the thing to be

proved. Q. E. F. quod erat faciendum, which was the thing to be done.

adj. . .adjacent. ad imposs.. ad impossibile, to an im

possibility. a fort.. .a fortiori, by a stronger

reason,

ext......

..

extr.....

bisd..

cen......

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alt..... . altitude. altr. .....

.alternate. antec ....antecedent. ang .........angular. assum.. .assumendo, by adopting. bis.,

.. bisects, or bisect.

.bisected. bisg.. .bisecting

..centre. ch........chord. com.......common. comp- ..compound. compl. ....complement. con. sup...contrary supposition. c. scr......

..circumscribe. c. scg. ... .circumscribing. conseq. ....consequent. cont.

.continued.. conterm ...conterminous. contn.......contain, or contained. descr..... describe, or described. descg. .describing. diag. diagonal.

.diameter. diff.

difference. dist.. .distance, or distant. div.

. , divide, or divided

dupl. ....duplicate. eq.. ... ...equal or equally. eq. ang.. .. equiangular. eq. lat.. .. .equilateral. ex, ab.. .. .ex absurdo, by an ab

surdity. ex gr..... .. .exempli gratiâ,

for example's sake.

.exterior, or exteriorly. extn.. externally.

.extremity, or extremities. fig.. ..figure. gr.. · greater. homol. .. homologous. hyp. .hypotenuse. incl.. .include, included. indef. .indefinitely inscr. inscribe, inscribed. int.. ... interior. inters. . intersect, intersection. intr ... .internal, internally. m or mult.

multiple magn. .magnitude. meas. mid.. .middle.

..obtuse. opp.. ..opposite. par.

.parallel. parlm. • parallelogram. pent. .pentagon. perp. • perpendicular.

.measure.

ob. .

diam..

sem. C.

qu. lat.

..

pos.. . position. prod. .... produce, produced. propl. .proportional. pt. .part. qu. ang. .. quadrangular.

.quadrilateral. rad. ..radius. rat.

.ratio. recip. reciprocal. rect. ...rectangle. rectl.. .. rectilineal. rectr.. .rectangular. reg.. .regular.

..remaining. resp. .. .. . respective.

rt. .. .. .. .right. sect.

..sector. seg. .segment.

..semicircle. sem. cirf....semicircumference. sim. .similar to, similarly. sim. sit. . .similarly situated. sq. . square.

..straight. suppl.. .supplemental. tang. .. tangent. rap.

.trapezium. undiv.. ...undivided. uneq.. .. unequal, or unequally.

... vertex, vertical.

st..

rem......

vert.. ....

GRADATIONS IN EUCLID.

BOOK III.

TREATING OF THOSE PROPERTIES OF THE CIRCLE, AND OF STRAIGHT

LINES IN AND ABOUT IT, WHICH CAN BE DEDUCED FROM

THE FIRST AND SECOND BOOKS.

A circle, strictly speaking, signifies the space bounded by a circumference, but in this book the term is employed sometimes to denote that space, and at other times, the circumference itself.

Euclid, too, occasionally assumes from experimental knowledge, certain properties of the circle, which a more rigid and exact method of reasoning would have established before using them. This is the case in the first Proposition itself, where it is taken for granted that the perpendicular to the chord of an arc will meet the circle in two points. In some instances also the method of indirect demonstration is adopted, when the more satisfactory method of direct proof is available; examples of this occur in Props. 2, 13, 16 and 36.

By restricting the meaning of the term angle to an opening formed by two conterminous lines, and less than two right angles,

B

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