Step by step man ascends the Himalayas and compasses the earth; step by step is the Mathematician's course. As in the First Part, so in this Second Part of the “ Gradations,” the same method has been pursued. Prefixed to the Proof are references to the principles that have to be employed, and often quotations of the very words in which Euclid embodies those principles. These are not so fully given, indeed, as in the First Part, for it is presumed that familiarity with leading principles and propositions has been already attained. Occasionally too, in the steps of the Construction and Demonstration, the special reference is not made in the margin to the evidence on which an argument or a conclusion rests ; but the Learner will scarcely find this any obstacle, if he has mastered what he has read.

It will be well for the Learner thoroughly to consider the references, before he proceeds to the Particular Enunciation, the Construction and the Demonstration of the Proposition ;-indeed, were he of himself to put together the truths with which he is supplied, and to see how the new truth is to be deduced from them, he would derive the best assistance, that from the reasoning of his own mind, to understand and appreciate the fuller proof of the formal demonstration. The Memory, no doubt, is a most valuable power in acquiring any kind of knowledge, but in Mathematics especially it is the understanding and the reasoning faculty that are employed to most advantage and developed with greater exactness.

As an instance of the method recommended, let the Learner take that important Proposition, 35, III, If two st. lines cut one another within a circle, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other.After weighing these words of the General Enunciation, let him call to


mind the several truths contained in the references under the heads, “ Con.," construction, and “DEM.," demonstration; and if he has forgotten any of them let him turn back to the very propositions numbered, as 10, I., 3, III., 47, I., &c., and carefully think them He thus burnishes

his old

weapons; and now let him try to trace out the connexion between the propositions referred to, and to ascertain how they lead to the new proposition which he seeks to establish. He will say to himself, here are several undoubted truths and facts presented to me; I have already accepted them as principles of Geometrical Reasoning,--and they are now given that I may demonstrate some other truth, or solve some other problem. Can I not, with the implements provided, build this new hou se and see how, like the others, it is composed of indestructible materials ?

He may rely, that by thus exercising his judgment, he will do more, than any mere effort of memory can do, for really understanding and retaining mathematical truths.

A very full Table of signs and abbreviations is given, and this should be consulted until they have become well known.

Some of the Demonstrations, as in 8, V, and 15, V, have been shortened. By the time the Learner has mastered so much of the Geometry, he will readily perceive the connexion of the argument, and not require the entire fulness of which it is capable; or, if he should, he

may be expected to supply it from his own resources. The Index was a subject of some consideration. An alphabetical Index had been prepared, but it was rejected, because it would have occupied too much space. The advantage of learners appeared to be more promoted by having the whole of the General Enunciations


of Euclid's Geometry brought together under the heads of Problems and Theorems, with their respective illustrations, applications, and

An alphabetical Index would have facilitated references to particular truths, but the consecutive or synoptical Index conduces more to the understanding of the whole work, and to the tracing out of the connexions of its parts.

A word or two to those who, from inexperience, do not understand the difficulty of avoiding errors of the press in a work where many signs, abbreviations, and references are used. As a most justly celebrated mathematician* has observed,—The Table of Corrigenda, at the end of the volume, “may convey an impression that the work is incorrectly printed, which is not the case;" and he adds, “If every mathematical work, at its completion, had the fruits of some years of examination presented to the reader, I know of none which would not have lists as large in proportion to their size and the number of symbols contained in them as the present one.” On this subject the author will simply remark, that should “the Euclid Practically applied" attain a second edition, these and some other faults will be carefully amended.

In his Mathematical Preface, “written at his poor House at Mortlake, Anno 1570, February 9," “ John Dee, of London," addressed himself 6 to the ynfained Lovers of Truthe and constant Studentes of Noble Sciences;" "he hartely wisheth them grace from heaven and most prosperous successe in all their honest attemptes and exercises." So, with him, I say to all who value good learning, “I commit you vnto God's Mercyfull direction for the rest; hartily beseechyng hym, to prosper your Studyes and honest Intentes to His Glory and the Commodity of our Country." October 1, 1861.

* DE MORGAN, in his Differential and Integral Calculus, A.D. 1842.

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1.-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.

v root. not greater than.

ratio. less than.

=: equality of ratios. * not less than.

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

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.




II.-Geometrical Signs. a point. *

A triangle. | straight line.

o parallelogram. parallel, parallel to.

square, Orectangle. Langle.

O circle.
I perpendicular to, at rt. Zs. ce 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 AR, or A B’, denote not the square of A, A B, but the square on A or A B.



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

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

III. - Additional Algebraic Expressions.

n or p


Magnitude. m multiple.

m+n 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

тип А

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.


Add .....

[blocks in formation]

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

Ènunciation of a Prop.

Example. Gen.

..... General Enunciation. H. or Hyp.Hypothesis of a Prop. H. 1 &c... Step 1 &c. of the Hyp. L ............Line. M. er Mag. Magnitude. P. or Prop. Proposition.

Proced ...Precedendo, by going on.
Prel. .Preliminary
Prod. Product.
Pst. or Psts.Postulate, or Postulates
Quæs. Quæsitum or Quæsita.
R... .Ratio.
Rec..... Recapitulation.
Remk. . . .Remark to be made.
Sch... Scholium or Scholia.
Sim.. ..So, similarly, by similar

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

dosition. Theor..... Theorem.


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