iv

vanced student.

The views which I have deve

loped have been adopted after much thought and mature consideration. If in If in any instance they are erroneous, I am confident that those who are best able to set me right will be most ready to pardon the error. In discussing the principles of mathematical science, it is impossible not to have come in collision with some of those who have previously written on them. With respect to authors of former days, it is sufficient to observe that I have not hesitated freely to canvass their opinions; such names as Barrow are too illustrious in the annals of literature to suffer from my strictures. Of the opinions of living authors I have spoken more guardedly. It has been my anxious desire to contend simply for truth; and although I have urged my own views with decision, I have felt all the diffidence which a proper respect for the judgment of others cannot fail to produce.

To the students (whether in Cambridge or in Edinburgh) who have attentively followed my Lectures, who have exhibited their interest in the

V

subjects discussed by so many voluntary essays, over and above the exercises exacted from them, -who have manifested towards me on all occasions the warmest reciprocity of affection; I cannot suffer this opportunity to pass without an expression of my gratitude for their kindness, and of my interest in their welfare and happiness.

The friend who has given his valuable and able assistance in conducting these sheets through the best thanks.

press

has my

EDINBURGH,

March 16. 1843.

P. KELLAND.

CONTENTS.

LECTURE I.

ON THE IMPORTANCE OF A CLEAR CONCEPTION OF FIRST

PRINCIPLES.

Introduction-Probable difficulties at the outset of science-illustrated

by optics-geometry introduced into Greece by Thales-originated

in Egypt-reply to Ritter-small amount of Egyptian geometry-

discoveries of Thales and Pythagoras-due to the acquisition of cor-

rect principles-their followers-problems of the quadrature of the

circle and duplication of the cube-conic sections, &c.-Euclid's

Elements unequalled-rapid progress of science when its principles

were established-reflections suggested by the subject,

LECTURE II.

Page 1

ON THE PRINCIPLES OF DEMONSTRATIVE GEOMETRY.

Geometry, the mainspring of sound education-subject-the properties

of space-how called up in the mind-externals suggestive their

agency neither to be confounded with the assistance they afford, nor

with the application which is made of geometry to them-space de-

fined-termination-extension-composition comparison-elements

of demonstration-definitions of two kinds-requisites for a good defi

nition-postulates-axioms of two kinds-principle of comparison by

superposition-the superiority of equality to congruity asserted-Pro-

clus defended against Barrow and Simson-equality the general of

which congruity is the particular mode of comparison in geometry, 22

LECTURE III.

ON THE DEFINITIONS, POSTULATES, AND AXIOMS OF

GEOMETRY.

Object of definitions different from that of axioms-without reference to

the nature of the thing no interpretation or extension possible

Locke's views need no refutation.-Definitions in Euclid's first Ele

ment-proof of the coincidence of planes-of the equality of right

angles-postulates-axioms-theory of parallels, dependent on our

idea of straightness-connection between the two axioms of a straight

line,

LECTURE IV.

ON RATIO AND PROPORTION.

Page 51

Measure-ratio-incommensurable magnitudes-proportion-Euclid's

doctrine how developed-arithmetical ideas-lead to a common pro-

perty-that property made the definition-necessity for the sole depen-

dence of the results on the conditions in order that an hypothesis and its

conclusion may be reciprocal-propriety of Euclid's definition-its ap-

plication-definition of inequality of ratios.

LECTURE V.

ON THE FOUNDATION OF ALGEBRA.

80

History-want of attention to the principles amongst many writers-ex-

ceptions, Peacock, Hamilton, De Morgan-my own views-definition

of algebra-a conception of the nature of the symbols and operations

requisite geometrical illustrations-imaginary geometry, and Reid's

geometry of visibles (note)—reasons for requiring an idea of operations

algebraic definitions of indices-signs-imaginary quantities—argu-

ments in support of the arithmetical derivation of algebra-difference

of two systems-reasons for preferring the arithmetical derivation-1°

in the other system risk is run that the conditions be inconsistent with

each other--2° a direct and reciprocal law have to be applied simul-

taneously-3° the transition to arithmetic as difficult as the transition

from it-conclusion,

LECTURE VI.

ON THE DIFFERENTIAL CALCULUS.

103

The Differential Calculus a subject which requires time for the mastery

of its principles-1. necessity for extension of methods of comparison

2. development of methods-that of Exhaustions of Indivisibles

of Infinitesimals-of Fluxions-prominent characteristic, the introduc-

tion of a subsidiary quantity-Residual Analysis-Derived Functions

-3. importance of notation-in arithmetic-in algebra-in science ge-

nerally-it indicates processes to the eye-extends them-classifies

forms and is a ready instrument for operation-4. methods actually

in use that of Limits the most to be recommended,

127