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purpose of teaching Geometry to large classes of pupils who are about entering upon the study of this important science; and of supplying a manifest defect now existing in our schools.

THOMAS FOULKE, Prin. W. School No. 14, N. Y.

We have examined the Chart of Geometry and " Book of First Lessons,” prepared by D. M'Curdy, and believe them to be well adapted to the purpose of teaching this science in public schools. In our opinion, a pupil, by using these charts, and aided by the teacher, will obtain a clearer idea of Geometry, besides saving time, than by any other method.

WM. KENNEDY, Prin: W. School No. 2, N. Y.
JOHN J. DOANE, Asst. W. School No. 2.

The undersigned, members of the Book Committee of the Ward School Teachers' Association, having examined a chart of the diagrams of Euclid, connected with a book called “ First Lessons,” and prepared for public schools by Mr. D. M'Curdy, respectfully

REPORT:

That the design of said book and chart appears to be to supply a defect in the elements of a sound practical education, the existence of which your committee must readily admit. The cause of this defect, however, is not in the want of books on Geometry; but in the oppressive strictness in which the subject is usually presented to beginners. The plan now presented obviates this difficulty by omitting the demonstrations, and requiring learners to read the propositions, recite the proofs and draw the diagrams, as occasional exercises, on slates. Questions are also suggested to teachers in view of the book and chart, which, if followed out, must inevitably render the subject familiar to both teachers and pupils; and prepare them to demonstrate the propositions of Euclid with facility and success.

Your committee therefore recommends this work to the patronage of the Association as worthy of its unanimous support, and offers the following resolution:

Resolved, That this Association recommend M'Curdy's geometrical chart of the diagrams of Euclid in connection with the book of “ First Lessons," with a view to their introduction into the Ward Schools of this city.

SENECA DURAND,

E. H. JENNEY,
Signed,

JOHN WALSH,
EDWARD M'ILROY.

At a regular meeting of the Ward School Teachers' Association on the 17th Dec., 1845, the above report of the committee was approved and the resolution unanimously adopted.

JOSIAH RICH, President W. S. T. Asso.

FURTHER DIRECTIONS FOR THE USE OF THE CHART OF

GEOMETRY AND BOOK OF FIRST LESSONS.

Respectfully submitted to the Officers and Teachers of the Public Schools.

GENTLEMEN, -In reply to a question proposed, How are the Chart and First Lessons, to be used ?--permit me to explain in a few words.

Each student requires a copy of the book; one chart will be sufficient for a large class. The students read the definitions, postulates, axioms and propositions, in order; and recite the proofs at each proposition, as directed. The teacher should not embarrass himself or his class, by exacting from them more than they can perform: he ought not to stop, in the first course, except to explain the meaning of terms. The object is to make the enunciations familiar to the students. The six books of Euclid and plane trigonometry occupy forty-three pages; these may be read and recited every six months, without any material abatement of other studies.

In the second course of reading, slates and pencils may be used :-in order to draw a straight line from one point to another—to produce the same in a straight line-and to describe a circle about a point with any radius. This is the permission sought in the postulates; and it embraces every thing that is to be done by hand.

Next, in connection with the definitions, the class draws straight lines, parallel, and at right and oblique angles to each other; also radii, diameters, and other chords in a circle;-triangles equilateral, isosceles and scalene-right angled and oblique ;-squares and other rectangles, rhombuses, rhomboids, and a variety of trapezia: then they should make several angles at the same point, and read them distinctly by their proper letters. In short, the class should draw every line and figure spoken of in the definitions, and call it by its proper name: for then they will take an interest in it as their own production.

In view of the axioms, every relation should be expressed by appropriate lines, angles, or areas; illustrating each according to the tenor of it: this exercise will impress the use of these simple propositions upon the learner's memory, and be to him a pleasing introduction to the process of ratiocination: axioms are always invincible proofs.

The teacher will now direct the class to copy from the chart as many of the diagrams as he thinks proper for the day's lesson : he will examine whether they have done it well or ill; and ask, in relation to each diagram, what proposition it illustrates, and what proofs are adduced ;--all of which the class will answer: and as they recite proof after proof, he will point to the proper diagram on the chart, and say—“This is the il. lustration-How does it apply to the case ?" He cannot expect answers in all cases: but the question will elicit attention.

The definitions and axioms are properly quoted as proofs: for although they are not separately represented on the chart, they are found all over it. The postulates prove only this, that the geometer asks permission to draw such lines as he requires: the fact of his drawing them shows that he has permission.

74

Some of the elements are of more general use in demonstrations than others, and are more frequently quoted. Of such is the equality of radii ;

of triangles identical (as in p. 4 of b. 1), and those upon the same or equal bases and between the same parallels; also of parallelograms in similar case;--of angles by position vertical and opposite, alternate, and in alternate segments of a circle made by a chord meeting a tangent, interior, exterior and adjacent;—of the opposite sides and angles of parallelograms; -of the square of the side subtending to the squares of the sides containing a right angle;--of the squares and rectangles upon lines divided equally and unequally, or bisected and produced ;-of ratios; and of similar rectilineal figures. These equals are among the most useful of the elements, and will claim the learner's attention.

The object of the First Lessons is to furnish the entire text, unencumbered with comment, which has ever precluded the general diffusion of these useful elements. The chart furnishes the diagrams, as an auxiliary to the acquirement of the text, which is the great desideratum. They do not speak from experience who say that students cannot learn the enunciations without the demonstrations: their doubts, however, will be received as arguments only by the indolent, “who stand by the brook until it shall have discharged all its waters.” On the other hand, no one will pretend to say, that the demonstrations can be effected without that familiarity with the propositions provided for in the Chart and First Lessons.

The time has arrived when the elements of geometry should obtain a place in the common education. Abstracts for the use of Surveyors, Navigators, Builders, &c., are insufficient: the qualification for an occupation, or office, should precede the exercise of it, and be the common portion of junior citizens. The old plan proves to be impracticable by the public destitution of this most useful knowledge: and it is by far too costly for large schools. The plan here proposed is accessible to every reader, and requires only the test of experiment. A book worth 25 cents, in the hands of willing teachers, will effect the general diffusion of these inestimable elements of natural science in the schools: not merely in view of the future occupations of the students; but as the basis of that general knowledge required by every citizen of the United States; as the best remedy for the contagion of the turbid streams now inundating town and country under the name of literature; which robs youth of its time for improvement, and leads many into the imitation of vices, at least palliated if not approved.

In regard to the manner of using the Chart and First Lessons, it would seem unnecessary to say more. The preface to the book contains forms of questions applicable to the diagrams and propositions. We shall add the substance of the foregoing directions. If any difficulty remain, it must be that of invincible incredulity: for what can be more simple than to read the book and understand what you can of it;--to repeat the same and learn more and more;-to follow out the directions given and make your knowledge perfect?

The readings and recitations are all the text of Euclid, without comment;-the collective wisdom of the past; by which the remote parts of the earth have been discovered, rendered habitable, and adorned with cottages, cities and temples to the praise of Him who reigns.

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OLMSTED'S ASTRONOMY: For the use of Colleges. By Denison Olmsted. 8vo. Sheep.

Price $200. Nearly all who have written treatises on Astronomy designed for teachers, appear to have erred in one of two ways; they have either disregarded demonstrative evidence and relied on mere popular illustrations, or they have exhibited the elements of the science in naked mathematical formulæ. The former are usually diffuse and superficial; the latter technical and abstruse.

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This work having been used as a class book in many schools and academies, the publishers have been favoured with testimonials of approbation, among which are the following: From Charles Henry Alden, Principal of the Philadelphia High

School for Young Ladies. I have examined with great care “ Olmsted's Compendium of Astronomy,"

,” and have taken a highly intelligent class in my institution critically through it. We have long felt the want of a Text Book in this most interesting science, and the author of this merits the thanks of the profession for a Treatise so entirely methodical and lucid, and so admirably adapted to our more advanced classes. Judging of its merits by the interest evinced by my pupils, as well as from its intrinsic excellence, I cannot too strongly recommend its general adoption.

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i have not vanity enough, I assure you, to suppose for a moment that any thing I can say will add to the fame of Professor Olmsted's Works on Natural Philosophy; but as you ask me my opinion of his

Compendium of Astronomy,” I will say, that I intend to introduce it irto my school, considering it the best work of the kind with which I am acquainted.

This work has also been very favourably noticed in various periodi cals. The following discriminating remarks from the New Haven Record, are understood to be from the pen of an able and experienced teacher of Astronomy.

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