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THE GEOMETRY OF PLANE TRIANGLES,
FOUNDED ON SIMSON'S TEXT;
SHOWING THE USES OF THE PROPOSITIONS, &c.;
“ GEOMETRY 13, PERHAPS, OF ALL THE PARTS OF MATHEMATICS, TIIAT WHICH OUGHT
Essais par s. F. Lacroix, p. 306.
MANCHESTER : JOHN HEYWOOD, 143, DEANSGATE.
LONDON : SIMPKIN, MARSHALL, & CO.
“How many a Common Artificer is there, in these Realmes,” “that dealeth with Numbers, Rule, & Cumpasses: Who, with their owne Skill and experience already had, will be hable (by these good helpes and informations) to finde out and deuise, new works, straunge Engines and Instrumentes; for sundry purposes in the Common Wealth ? or for private pleasure ? and for the better maintayning of their owne estate ?”
JOHN DEE, His Mathematicall Preface, A.D. 1570
GEOMETRY, land-measuring, as the word denotes (from gee, earth, or land, and metron, a measure), was in its origin an Art and not a Science: it embraced a system of rules, more or less complete, for performing the simpler operations of land-surveying; but these rules rested on no regularly demonstrated principles, they were the offspring rather of experiment and individual skill, than of scientific research.
The points and lines of such a Geometry were necessarily visible quantities. A mark, which men could see, would be their point; a measuring rod, or string, which they could handle, their line ; a wall, or a hedge, or a mound of earth, their boundary. The first advance beyond this would be to identify the instruments which they used in measuring, with the lines and boundaries themselves; the finger's breadth, the cubit, the foot, the pace, would become representatives of a certain length without reference to the shape. It was only as the ideas and perceptions of those who cultivated the art of measuring grew more refined and subtile, that an abstract Geometry would be evolved, such as Mathematicians understand by the term, in which a point marks only position; a line, extension from point to point; and a surface, space enclosed by mathematical lines.
Geometry thus understood, has been defined in general terms to be, “the Science of Space.” It investigates the properties of lines, surfaces, and solids, and the relations which exist between them. Plane Geometry investigates the properties of space under the two aspects of length and breadth; Solid Geometry, under the three,-of length, breadth, and thickness. It is the consideration of the Elements of Plane Geometry on which we are about to enter.
According to Proclus, EUCLID of Alexandria flourished in the reign of the first Ptolemy, B.C. 323-283. To him belongs the glory, for such it is, of having colleeted into a well-arranged system, the scattered principles and truths of Geometry, and of having produced a work, which, after the test of above twenty centuries, seems destined to remain the Standard Geometry for ages to come.
Euclid's work comprises thirteen books, of which the first four and the sixth treat of Plane Geometry; the fifth, of the Theory of Proportion, applicable to magnitude in general; the seventh, eighth, and ninth, are on Arithmetic; the tenth, on the Arithmetical Characteristics of the divisions of a straight line; the eleventh and twelfth, on the Elements of Solids; and the thirteenth, on the Regular Solids. To the thirteen books by Euclid, Hypsicles of Alexandria, about A.D. 170, added the fourteenth and fifteenth books,-also on the Regular, or Platonic Solids.
Of the Six Books, the first may be described in general terms, as treating of the Geometry of Plane Triangles; the second, of Rectangles upon the parts into which a straight line may be divided ; the third book, of those Properties of the Circle which can be deduced from the preceding books; the fourth book, of such regular and straight-lined figures as can be described in or about a circle; the fifth, of Proportion with regard to magnitude in general; and the sixth, of Similar figures, and of Proportion as applied to Geometry.
For an outline of the origin and progress of the science of Geometry, the learner should consult the Introduction to the Elements of Euclid, edited by Robert Potts, M.A., Trinity College, Cambridge.
It is of great advantage to have stored in the memory the very words of the Definitions, Postulates and Axioms, and of the more important Propositions; and associate with the words the numbers, as Definition 15, Axiom 8, Proposition 4, 8, 26 &c., of book I.
But in the construction of Geometrical figures, and in the demonstration and application of Geometrical Truths the Reasoning Faculty should be chiefly employed. A youth may repeat cleverly by rote, and yet be ignorant of the principles of the science. In the study of Geometry no aids are so effectual as the determination and the endeavour thoroughly to understand the process of reasoning, and the nature and force of the argument.
As an instance of the method recommended to the learner, let him take that important Proposition, the 32nd of book I. Having well considered the meaning of the words, let him, by reference to Prop. 31, and also to Prop. 29, Ax. 2, Prop. 13, and Ax. 1, recall to mind what is required for the construction of the figure, and what for the deinonstration of the theorem. He will say to himself, here are several undoubted truths and facts;-) have already proved them and accepted them as principles of Geometrical Reasoning; and they are now given me that I may arrive at other truths. By using them, can I not demonstrate the inference or conclusion of the proposition?
He may rely that by thus exercising his judgment, he will do more than by any mere effort of memory, for really understanding and retaining Mathematical truths.
For fuller information learners are referred to the Author's larger work, Euclid's Plane Geometry, Books, I-VI., Practically Applied, Parts I-II.
SIGNS AND ABBREVIATIONS.
Signs in Geometry possess equal accuracy with words, and far greater clearness than the undivided paragraphs of Simson, or even than the broken clauses of Potts. Besides, “ Cambridge has allowed the use of certain symbols in Geometry before forbidden by her;" and Lacroix long since declared, in 1805, “ The History of Mathematics proves, that it is the more and more extended use of arbitrary symbols, contrived with the purpose of abridging expressions, or of rendering their analogy evident, which has most contributed to the progress of the science, by relieving the memory, and by facilitating the combinations of the given relations and reasonings.” See “Essais sur l' Enseignement &c.,” p. 227.
One strong recommendation or them has thus been pointed out; "It quite possible, and, in fact, frequently happens, that a boy gets up his Euclid lesson by rote from the ordinary editions now in use;" but, “it is almost, if not quite impossible to bring the memory simply into play when a proposition of Euclid is set forth in symbolic language.” The reasoning powers must be exercised, or the work will not be accomplished.
I.- Arbitrary Signs, common to Arithmetic and Algebra, and often used in Geometry. because. equals, or cqual.
by, divided by. therefore. + not equal to.
v root. greater than.
+ plus, more, increased by. ratio. not greater than minus, less, lessened by. :=: equality of ratios. less than.
w difference between. ::: : proportion. not less than, X into, multiplied into. : : : progression.
II.--Representative, or Geometrical Signs. a point. Langle.
square. O rectangle. 1 straight line. I perpendicular to. O parallelogram. il parallel to. A triangle.
circle. Oce circumference. A single capital letter, as A, or B, in reference to a Diagram, denotes the point X, or the point B.
Two capital letters, as AB, or CD, also in reference to a Diagram, denote the straight line AB, or CD; but when the two letters indicate opposite angles, they denote a square, a rectangle, a parallelogram, or a polygon, as the figure will show.
A capital letter and numeral, as A?, or two capital letters and numeral, as AB”, denote the square on the straight line A, or on AB.
III.-Abbreviations. Add. Addendo, by adding. E. Exposition.
Sim. Similarly. App. Application. H. or Hyp. Hypothesis. S. or Sol. Solution. Ax. Axiom.
P. or Prop. Proposition. Sub. Subtrahendo, by Conc. Conclusion. Prob. Problem.
taking away. Cor. Corollary. Pst. Postulate.
Sup. Suppose. C.
Construction. Quæs. Quæsitum, or-a. Superp. SuperponenDat. Datum, or data. Rec. Recapitulation. .do, by superposition. Def. Definition, Remk. Remark.
Theor. Theorem. D.
Demonstration. Sch. Scholium. 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. ad imp.
ad impossibile, reduced to an impossibility. a fort. a fortiori, by a stronger reason.
assumendo, by assuming, or adopting.
ex absurdo, by an absurdity. alt. alternate. int. interior.
rem. remaining. com. common. opp. opposite.
rt. right. con, sup. contrary sup- par. parallel.
sq. square. position. quadrang. quadrangular, st. straight. equiang. equiangular. quadril. quadrilateral. uneq. unequal. equil. equilateral. rectil. rectilineal. vert. vertex. ext. exterior.