Isaac Todhunter

Name: The Elements of Euclid for the Use of Schools and Colleges
Author: Isaac Todhunter

The Elements of Euclid for the Use of Schools and Colleges (1872)

Summary

Isaac Todhunter's The Elements of Euclid for the Use of Schools and Colleges is an educational adaptation of Euclid's foundational geometric treatise, structured to clarify principles of plane geometry through rigorous logical reasoning. Covering topics such as angles, circles, parallelograms, and proportionality, the text systematically introduces postulates, definitions, and theorems, highlighting the relationship between construction and demonstration.

Todhunter’s work highlights the constraints of classical geometry, as seen in discussions of unsolvable problems like squaring the circle, while upholding Euclid’s methodical approach to spatial relationships. This edition functions as a pedagogical link between ancient mathematical theory and contemporary analytical precision.

Quotes from The Elements of Euclid for the Use of Schools and Colleges (Isaac Todhunter)

Isaac Todhunter, The Elements of Euclid for the Use of Schools and Colleges (1872)

“ That a proposition may be false and yet furnish consequences which are true, can be seen from a simple example. Suppose a theorem were proposed for investigation in the following words; one angle of a triangle is to another as the side opposite to the first angle is to the side opposite to the other. If this be assumed to be true we can immediately deduce Euclid's result in I. 19; but from Euclid's result in I. 19 we cannot retrace our steps and establish the proposed theorem, and in fact the proposed theorem is false. ”

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Source: Wikisource

Isaac Todhunter, The Elements of Euclid for the Use of Schools and Colleges (1872)

“ There are three famous problems which are now admitted to be beyond the power of Geometry; namely, to find a straight line equal in length to the circumference of a given circle, to trisect any given angle, and to find two mean proportionals between two given straight lines. The grounds on which the geometrical solution of these problems is admitted to be impossible cannot be explained without a knowledge of the higher parts of mathematics ”

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Source: Wikisource

Isaac Todhunter, The Elements of Euclid for the Use of Schools and Colleges (1872)

“ The definition of a square assumes more than is necessary. For if a four-sided figure have all its sides equal and one angle a right angle, it may be shewn that all its angles are right angles; or if a four-sided figure have all its angles equal, it may be shewn that they are all right angles.
Postulates. The postulates state what processes we assume that we can effect, namely, that we can draw a straight line between two given points, that we can produce a straight line to any length, and that we can describe a circle from a given centre with a given distance as radius. ”

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Source: Wikisource

Isaac Todhunter, The Elements of Euclid for the Use of Schools and Colleges (1872)

“ When there is more than one hypothesis or more than one conclusion to a proposition, we can form more than one converse proposition. For example, as another converse of I. 5 we have the following: if the angles formed by the base of a triangle and the sides produced be equal, the sides of the triangle are equal; this proposition is true and will serve as an exercise for the student. ”

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Source: Wikisource

Isaac Todhunter, The Elements of Euclid for the Use of Schools and Colleges (1872)

“ The inclination of a straight line to a plane is the acute angle contained by that straight line, and another drawn from the point at which the first line meets the plane to the point at which a perpendicular to the plane drawn from any point of the first line above the plane, meets the same plane.
6. The inclination of a plane to a plane is the acute angle contained by two straight lines drawn from any the same point of their common section at right angles to it, one in one plane, and the other in the other plane. ”

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Source: Wikisource

Isaac Todhunter, The Elements of Euclid for the Use of Schools and Colleges (1872)

“ Lastly, we have the demonstration itself, which shews that the problem has been solved, or that the theorem is true.
Sometimes, however, no construction is required; and sometimes the construction and demonstration are combined.
The demonstration is a process of reasoning in which we draw inferences from results already obtained. These results consist partly of truths established in former propositions, or admitted as obvious in commencing the subject, and partly of truths which follow from the construction that has been made, or which are given in the supposition of the proposition itself. ”

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Source: Wikisource

Isaac Todhunter, The Elements of Euclid for the Use of Schools and Colleges (1872)

“ In a circle the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.
Let ABCD be a circle, of which BC is a diameter and E the centre; and draw CA, dividing the circle into the segments ABC, ADC, and join BA, AD, DC: the angle in the semicircle BAC shall be a right angle; but the angle in the segment ABC, which is greater than a semicircle, shall be less than a right angle ”

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Source: Wikisource

Isaac Todhunter, The Elements of Euclid for the Use of Schools and Colleges (1872)

“ Again, the locus of the vertices of all triangles which are on the same base and on the same side of it, and which have the same vertical angle, is a segment of a circle described on the base; for it is shewn in III. 21, that all the points thus determined satisfy the assigned conditions, and it is easily shewn that no other points do.
We will now give some examples. In each example we ought to shew not only that all the points which we indicate as the locus do fulfil the assigned conditions, but that no other points do. ”

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Source: Wikisource

Isaac Todhunter, The Elements of Euclid for the Use of Schools and Colleges (1872)

“ If any point be taken without a circle, and straight lines be drawn from it to the circumference, one of which passes through the centre; of those which fall on the concave circumference, the greatest is that which passes through the centre, and of the rest, that which is nearer to the one passing through the centre is always greater than one more remote; but of those which fall on the convex circumference, the least is that between the point without the circle and the diameter; and of the rest, that which is nearer to the least is always less than one more remote ”

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Source: Wikisource

Isaac Todhunter, The Elements of Euclid for the Use of Schools and Colleges (1872)

“ In offering to students and teachers a new edition of the Elements of Euclid, it will be proper to give some account of the plan on which it has been arranged, and of the advantages which it hopes to present.
Geometry may be considered to form the real foundation of mathematical instruction. It is true that some acquaintance with Arithmetic and Algebra usually precedes the study of Geometry ”

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Source: Wikisource

Isaac Todhunter, The Elements of Euclid for the Use of Schools and Colleges (1872)

“ THEOREM.
In any right-angled triangle, any rectilineal figure described on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle. ”

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Source: Wikisource

Isaac Todhunter, The Elements of Euclid for the Use of Schools and Colleges (1872)

“ If in a circle two straight lines cut one another, which do not both pass through the centre, they do not bisect one another.
Let ABCD be a circle, and AC, BD two straight lines in it, which cut one another at the point E, and do not both pass through the centre: AC, BD shall not bisect one another.
If one of the straight lines pass through the centre it is plain that it cannot be bisected by the other which does not pass through the centre. ”

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Source: Wikisource

Isaac Todhunter, The Elements of Euclid for the Use of Schools and Colleges (1872)

“ If two straight lines meeting one another he parallel to two others that meet one another, and are not in the same plane with the first two, the first two and the other two shall contain equal angles. ”

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Source: Wikisource

Isaac Todhunter, The Elements of Euclid for the Use of Schools and Colleges (1872)

“ The volume of a prism is found by multiplying the number of square inches in its base by the number of inches in its altitude; the volume is thus expressed in cubic inches. Or we may multiply the number of square feet in the base by the number of feet in the altitude; the volume is thus expressed in cubic feet. By the base of a prism is meant either of the two equal, similar, and parallel figures of XI. Definition 13; and the altitude of the prism is the perpendicular distance between these two planes. ”

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Source: Wikisource

Isaac Todhunter, The Elements of Euclid for the Use of Schools and Colleges (1872)

“ And universally, if a straight line contain a unit of length an exact number of times, the area of the square described on the straight line is denoted by the square of the number which denotes the length of the straight line. ”

Isaac Todhunter

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Quotes from The Elements of Euclid for the Use of Schools and Colleges (Isaac Todhunter)

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