tions and applications of Geometry as a lowering of the dignity of the science.* Something of this feeling has been associated with the subject from the days of Euclid almost to the present time. Experience and reflection alike show that Geometry ought not to be introduced to the student in a purely abstract form. The inductive process by which generalizations and abstractions are first acquired, is usually the best for communicating them * “ The ancient philosophy disdained to be useful, and was content to be stationary. It could not condescend to the humble office of ministering to the comfort of human beings. Once, indeed, Posidonius, a distinguished writer of the age of Cicero and Cæsar, so far forgot himself as to enumerate, among the humbler blessings which mankind owed to philosophy, the discovery of the principle of the arch and the introduction of metals. This eulogy was considered as an affront, and was taken up with proper spirit. Seneca vehemently disclaims these insulting compliments. Philosophy, according to him, has nothing to do with teaching men to rear arched roofs over their heads. The true philosopher does not care whether he has an arched roof or any roof. Philosophy has nothing to do with teaching men the uses of metals; she teaches us to be independent of all material substances, of all mechanical contrivances. The wise man lives according to nature. Instead of attempting to add to the physical comforts of his species, he regrets that his lot was not cast in that golden age when the human race had no protection against the cold but a cavern. To impute to such a man any share in the invention or improvement of a plough, a ship, or a mill, is an insult. In my own time,' says Seneca, 'there have been inventions of this sort, transparent windows, tubes for diffusing warmth equally through all parts of a building. But the inventing of such things is drudgery for the lowest slaves ; philosophy lies deeper. It is not her office to teach men how to use their hands. We shall next be told that the first shoemaker was a philosopher.'”-Macaulay's Essay on Lord Bacon. to the young and uninitiated. The mingling together of theoretical deductions and practical applications, not only makes the subject more interesting, but also assists the student in grasping the doctrines of pure Geometry. Neither is it advantageous to teach Practical Geometry and Mensuration by prescribing bare constructions in geometrical drawing, and rules for the measurement of lines and surfaces, without demonstration or discussion. Consequently, in this text-book fundamental notions are deduced from concrete illustrations; the demonstrations are of the simplest character, and are followed by explanations of the use of the propositions in the ornamental and industrial arts. At the end of each chapter are given questions on the text, and on the application of Arithmetic to Geometry, with theorems and problems for solution. By means of this plan the study of Geometry may be commenced at a much earlier age than has hitherto been possible. The conception of moving points and lines, which gives to many proofs simplicity and clearness, is frequently employed, as also the method of superposition, which in a number of cases affords a concise and elegant demonstration. The constructions are such only as are practically useful, and thus a great impediment to progress is removed from the path of the pupil. The doctrine of proportion, one of the most important in Geometry, is very early expounded ; and by its simplification of the course of Plane Geometry, access to the Solid is afforded to many who otherwise would never obtain any useful knowledge of it. The logical relations of the propositions to one another, and the more difficult questions, such as the application of proportion to incommensurable magnitudes, are not discussed in this elementary treatise, as they naturally belong to an advanced or collegiate course. PREFACE TO SECOND EDITION. . The first impression having been exhausted, the present edition of Plane Geometry has been carefully revised, and a number of demonstrations and constructions modified in accordance with suggestions made during the recent discussions on Geometrical instruction, and by teachers who have used the work. Solutions of important problems have been added, together with new woodcuts wherever necessary, a recapitulation of the principal enunciations of Plane Geometry and Mensuration, and an extended index. To save the time of teachers, the author has prepared solutions of the exercises, which, though concise, will be found to contain all that the student needs for those examinations at which a knowledge of modern methods is desirable. Straight and Curved Lines.- Plane Surfaces, and their Intersection. Horizontal and Vertical Lines.-Foot-rule.-Measurement of Lines. -Standards of Length:-Tape Measure.-Compasses.-Surveyor's Definitions of various Angles ; how named.-Perpendiculars and Obliques.--Angles formed by intersecting Lines.-Set Squares.- Geometrical Locus.-Symmetrical Figures. -Axis of Symmetry. -. T Square.-Water-level.-Propositions on Perpendiculars and Obliques.-Mason's Level.-Bevel. Definitions of Circumference, Centre, Radius, Diameter.-Construc- tion of Circles.-Sliding Gauge.--Arcs.--Degrees, Minutes, and Seconds.-Ratios of Magnitudes.-Chords.- Propositions on equal Circles.--Calibers. - Semicircle.--Quadrant.-Expression of An- gles at centre in Degrees.- Protractor.-Construction of Equal Definitions.-Relations between the sides and Angles.-Geometrical Designs.—Construction of Triangles. Properties of Parallels.-How laid down in Plans and in Surveying. -Parallels applied to solve Questions on Triangles, and to construct Parallelograms.-Carpenter's Gauge.- Quadrilaterals defined and examined.—Diagonal. – Trisection of Angles.-Rhombus.-Rect- angle. --Square. -IsoscelesTrapezoids in woodwork.-Parallelogram and Diagonal in Mechanics.-Rectangular Pavements. Equivalence of Area (Parallelograms and Triangles). -Squares on Sides of Right-angled Triangle.-Measurement of Areas.-Unit of Surface. - Relation between Rectangles formed by Segments of a Line. - To find Areas of regular or irregular Figures.-Cross-staff. -Field-book ; Offsets, Base-line.-To measure horizontal and vertical Distances between Points.-Levelling-staff, Back-sight, Numerical Proportion.-Definition of Term, Extreme, Mean. --Pro- portion of straight Lines and Areas ; their Relation to Proportion of numbers.- Intersection and Interception of Lines proportionally:- Diagonal Scale.—To find Proportionals. -Reduction Scale.-Pro- portional Compass. -Pantograph.-Measurement of inaccessible CHAP. VIII.-Circles and Secants Relations between Circles and Straight Lines.-Division of Angles.- Angles at Centre and Circumference.-Properties and Relations to each other of various Secants and Segments of Secants; their Application to the Construction of Machinery. Properties and Construction of Tangents.—_ angents common to two Circles.-Applications of the Tangent.--Relations of Chords, Seg- ments, Arcs, and Angles, to the Tangent. --Solution of Problems on CHAP. X.-Combinations of Circles . Concentric Circumferences.-Construction and Properties of Circles touching or cutting one another.-Inscription and Description of equal Circles touching one another:-Connection of Wheels, Spindles, etc., in motion.-Ovals and Elliptical Figures, how derived. -Archi- tectural applications.-Ionic Volute.-Gothic Arches and Windows, Conditions of Similarity.-Homologous Sides.-Ratio of Similitude.- Problems and Constructions deduced from Similarity.-Relation of Areas of Similar Figures.-Equivalence and Similarity contrasted. Terms applied to Polygons.-Regular Polygons defined; Deductions from their Properties.-Circles inscribed in and described about Polygons.—Opposite Vertices and Sides, and Symmetrical Axes of Polygons. --Auxiliary Curve.-Graduation of Circles.-Grapho- meter. -Polygons in Art.--Mosaics.-The Involute to the Circle. Polygons, constructed by Reduction or Extension. CHAP. XIII.-Equality and Similarity of Polygons . 232 Regular Polygons investigated in same Manner as with other Figures. -Equality by Symmetry.—Division of Polygons.-Portable Table for taking Plans.- Problems on Circles, deduced from many-sided CHAP. XIV.-Miscellaneous Problems Problems on Triangles and Rectangles. To find the Area and Height of Triangles when the Sides are given. RECAPITULATION OF PRINCIPAL THEOREMS AND PRO- |