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An opinion is widely entertained, namely, that algebra should have priority of geometry in the order of study. This reverses the natural order of the studies; for what is algebra but a method of managing arithmetic and geometry ?
It prepares certain general formulæ, and teaches the reduction of them, by transposition, substitution, elimination, &c. This may be done as an envelope is prepared for a letter :—the letter must be enclosed, or the envelope is of no use. It requires knowledge of the relations of numbers, of magnitudes, or figures, as the case may be, to dispose in an equation the data and quæsita of a proposition. Any simple problem will illustrate the absolute dependence of algebra upon geometry, and how preposterous the idea of giving it priority. Let us seek the ordinate of a circle from its relation to the diameter (d) and the abscissa (x): the formula is this, v dx-x=ordinate. Now, in view of the diagram, these relations are plainly seen; but without it, the formula would be as abstruse to the juvenile capacity as the Chinese language.
After this manner, a multitude of useful theorems are lost to the community, from two causes: one is the neglect of geometry; the other is, that the theorems are placed under the unfriendly umbrage of algebraic symbols; and this latter calls itself the “modern improvements of science !" But that is no improvement in science which precludes the general diffusion of knowledge. There is therefore a want of order, as well as a defect, in the parts which constitute the elementary education.
Mathematical reasoning is conducted according to two methods ; one is called the method of analysis, or resolution ; the other is called the method of synthesis, or composition. Algebra adopts the former of these ; separating the known from the unknown parts of a general proposition; representing number and magnitude by symbols, and descending, by a succession of equivalent propositions, from the most complex to the simplest form. Geometry adopts the synthetic method, which begins at the simplest elements, and proceeds, by easy steps, to the more complex combinations; and it is proper to remark, that this alone is the process by which the known and the unknown parts of a general proposition can be distinguished.
It is obvious that the cost of books, on this plan, and
the labor of the study, are both greatly reduced; and the method by previous recitations of the text, which is the exclusive object of the book of "First Lessons,” and which is continued in the present volume, will enable teachers less proficient to use the work, without apprehension of error or loss of time. There is, moreover, an assurance of success connected with this system, which no other has given, or can give. What is the deficiency with all of us ? Why have we not more men entitled to de. grees? The reason is this: We are not masters of the elements of science; we cannot call up at will the proofs of many propositions: we wanted this culture when we were a young people; we want it still, and should take heed lest we entail the same want on our posterity.
In allusion to the peculiarities of this work, it is unne. cessary to be specific. Let the book be examined on its merits, and candidly compared with volumes of two or three times the size; no defect, it is believed, will be found in it, and it will not seem to be redundant. The order of Euclid has been preserved, because it has never been excelled; but the repetitions which swell other edi. tions and perplex the learner, are here obviated by the plan of previous recitations.
The demonstrations of the fifth book have been simplified exceedingly; and in one or two instances reduced from two octavo pages to a few lines: not by the substitution of symbols for words, but by a new definition of ratio, and a slight alteration in the method of compounding ratios. Nevertheless, care has been taken to introduce nothing which could not be directly employed in drawing out the properties of proportionals according to the rigor of the Euclidian geometry. The change consists in fixing definitely the value of ratio agreeably to its general use in the mathematics.
Mental arithmetic and mental algebra are sublimated abstractions, not affording the proper exercise for the juvenile mind. Children are entitled to the use of their senses before they are required to reason : because, to reason is to combine, compare, digest, and dispose in order the inaterials received through the senses.
Now, arithmetic furnishes an infinite variety of series, the terms of which have varied relations to each other:
but the law of each series must be known, in order to bring those relations into view. The relations of geometry, however, are presented to the eye;—they are realities; the business of the world in miniature: it reasons from certain data, and furnishes the best model of reasoning in things less certain.
It was not design that brought arithmetic to occupy the exclusive ground it has held in the schools. Children could always repeat several terms of the natural series of numbers, and merchants could always sum and sever their gains and losses: in addition to this, the Moorish system of notation had come into use, carrying the series beyond any assignable limit. At the time of the revival of letters, a basis was thus provided for the study of arithmetic; and to read, write, and cipher, was esteemed accomplished scholarship. No regard was had to the mechanical operations of the world, or to the laws of the universe. Since that time, arithmetic has been compiled and re-modelled into a thousand forms, and algebra superadded; but geometry, which should be, at least, co-ordinate with arithmetic, has been neglected, or left to the speculative scholar. Yet these elements belong to the operatives—to the employments of men; they arm men with the skill and force of Nature, and imbue them with her wise designs; they verify the adage that "knowledge is power."
There is a solemn voice in the natural truths of geometry, which calls upon School Officers, Professors, and Teachers, to sow the seed, to scatter it broadcast over this national husbandry, to sacrifice the distinctions of learning to the perpetuation of a wise popular government, through the medium of an efficient elementary education. And this voice would be obeyed, and this motive deemed sufficient, if something great were required to be done; but when it is merely to put into the hands of children the text of Euclid, to be read and recited, will they not say,
" Where is the use of it ?” Where then is the use of scattering so much good wheat over the fields ? Is it all lost? No? the stoutest doubter expects twenty to one of the same kind. By the same rule, therefore, we may have twenty Euclids to one from the Common Schools and Academies.
New York, March, 1846.
SECOND LESSONS IN GEOMETRY.
Definitions. 1. GEOMETRY is the science which treats of the similarity, equality, difference and proportions of magnitudes and of figures of extension.
2. A point is a position, or station in a line, at the extremities of a finito line, also at the meeting and intersection of lines: but it is not the measuring unit, nor any part of the measure of a line.
Cor. Hence, points are by position, central, angular, sectional, or extreme.
3. A finite line is that of which the extreme points are given.
Note. There are two classes of lines; namely, straight and curved: of curves there are several species; but the circle alone will be here considered. Lines have lengths, but no other dimensions.
4. A straight line is the path of a point, without curve or anglo.
Cor. Two straight lines cannot meet and part and meet again: thoy cannot have a common segment; but, meeting in two points, or coinciding in part, they shall coincide in all their length.
Note. Straight lines have certain relations to one another from their position ; namely, perpendicular, meeting, insisting, parallel, and intersecting. One line is perpendicular to another when it makes the adjacent angles equal, or when it pends, or hangs upon the other as the plumbline upon the level : lines meet when they touch and do not cut one another; one line insists upon another when it stands upon a point in the other: one line is parallel to another when it is in the samo plane with the other, and at all points equidistant from it: lines intersect one another when they pass through the same point.
5. A circular line is the path of a moving point about a stationary one, at the same uniform distance from it.
6. A superficies is the upper or outside face-the surface: it has two dimensions-length and breadth: it is bounded by a line, or lines ; and the intersection of two superficies is a line.
7. A plane is a superficies described by the lateral motion of a straight line; or, by its rotary motion about one of its extreme points.
8. A plano angle is the rotary declination of one straight line from another, about a stationary point in which they meet.
Note. Angles are, by position, adjacent or opposite, interior or er. terior, vertical or alternate ; by magnitude, they are acute, right, or obtuse ; the acute and obtuse are called oblique angles. Salient and re
entrant angles are the outward angles in fortifications; the former are greater and the latter less than half the compass of the angular point.
9. An acute angle is any declination of two straight lines, smaller than one-fourth of the compass of the angular point.
10. A right angle is the declination of two straight lines to one-fourth of the compass of the angular point.
11. An obtuse angle is any declination of two straight lines greater than one-fourth but less than half the compass of the angular point.
Cor. When the declination of two straight lines is equal to half the compass of the point in which they meet, they form no angle, but are in one straight line.
12. A plane figure is any form of a superficies,—it is bounded by at least three straight lines.
13. A circle is a plane figure enclosed by a uniformly curved line, called the circumference.
14. The centre of a circle is a point within it, equidistant from every point of the circumference.
15. A radius is any straight line drawn from the centre to the circumference of a circle: therefore all radii of the same or equal circles are equal to one another.
16. A diameter of a circle is a straight line drawn through the centre to the circumference on either side ;-of a parallelogram is that straight line which joins opposite angles.
17. A semicircle is a figure contained by the diameter and half the circumference.
18. An arc is any part of the circumference of a circle,-a chord is the straight line which joins the extremities of the arc.
19. A sector is the figure contained by two radii and an arc.
20. A segment of a circle is a part cut off by a chord,-a segment of a line is a part cut off or distinct.
21. Rectilineal figures are those which are enclosed by straight lines : trilaterals have three sides ; quadrilaterals, four, &c.
22. Trilateral figures, or triangles, have six parts ; namely, three sides and three angles ; from which they take the following names :
23. An equilateral triangle has three equal sides.
29. Quadrilateral figures, or quadrangles, are contained by four straight lines: of this kind are the square and oblong, the rhombus and rhomboid, the trapezoid and trapezium.
30. A square has four equal sides, and four right angles.
31. An oblong, or rectangle, has four right angles, and its opposite sides equal and parallel.
32. A rhombus has four equal sides, of which the opposite are parallel; and four oblique angles, of which the opposite are equal.
33. A rhomboid has its opposite sides and angles equal, and all its angles oblique.
34. A trapezoid has two of its opposite sides parallel.