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respectively the number of straight lines and the number of circles drawn.

To draw a line,

FUNDAMENTAL CONSTRUCTIONS

1. at will, Op. (R2).

2. through a given point, Op. (R1+R2).

3. through two given points, (Op. (2R1+R2). To take a given length in the compass, Op. (2C). To draw a circle,

1. at will, Op. (C3).

2. with an indefinite radius but with a given center, Op. (C1+C3), 3. with a given center and which passes through a given point, Op. (2C1+C3)

4. with an indefinite center but with a given radius, Op. (2C1+C3). 5. with the radius a given length and the center a fixed point, Op. (3C1+C3)

To lay off on a line which is drawn a length from an undetermined point of this line, or starting from a point on this line, the length comprised between the arms of the compass. Op. (C2+C3), or Op. (C1+C3).

ELEMENTARY CLASSICAL PROBLEMS

1. To construct a right angle, or to draw two perpendicular lines.

(a) Draw a circle (C3); a line cutting the circle in A and B (R2). Connect A and the center O of the circle (2 R1+R2). OA cuts the circle again in C. Draw CB (2R+R2); the angle CBA is a right angle. Op. (4R1+3R2+C3). Simplicity 8. Exactness 4. 3 lines, 1 circle.

(b) Draw any two circles O(r), O'(r), intersecting in A and B (2C3). Draw AB and OO' (4R1+2 R2). Op. (4R1+2R2+2C3). S. 8; E. 4. 2 lines, 2 circles.

(c) Draw a line (R2); with any two points of this line as centers draw circles (2C2+2C3) intersecting in A and B; draw AB (2R1+R2) Op. (2R1+2R2+2C2+2C3). S. 8; E. 4. 2 lines, 2 circles.

II. To find the length of the radius of a circle of which the center is not given.

P being an arbitrary point on the circle, draw any circle P(r), (C2+C) which cuts. the given circle in A and B. Draw B(r) (C+C3) which cuts P(r) in two points. Join either one of them, say C, to A (2R,+R2) AC cuts the given circle in D. DC is the length of the radius.

Op. (2R1+R2+C1+C2+2C3) S. 7; E. 4.

1 line, 2 circles.

D

III. At a point C on a given line AB, to erect a perpendicular to this line. (a) The classical construction has for its symbol Op. (2R,+R2+3C1+3C3). S. 9; E. 5. 1 line, 3 circles.

(b) Geometrographic construction. Place one point of the compass in an arbitrary point O, place the other on C (C1), draw O (OC) (C3) which cuts CA in A. Draw AO (2R1+R2) cutting O (OC) in D; draw DC, (2 R1+R2). Op. (4R1+2 R2+C1+C3) S. 8; E. 5. 2 lines, 1 circle.

IV. Through a point A to draw a parallel to the line BC. (a) and (b) the two constructions commonly given have the following symbols, Op. (2R,+R2+5C1+3C3). S. 11; E. 7. 1 line, 3 circles.

Op. (3R1+2R2+5C1+3C3). S. 13; E. 8. 2 lines, 3

circles.

(c) Draw A (r) giving B, then B (r) giving C (2C+2C). Draw C (r) which cuts A (r) in D (C,+C3). Draw AD (2R1+R2).

Op. (2R,+R2+3C1+3C3). S. 9: E. 5. 1 line, 3 circles. (d) Draw a circle O (OA) (C1+Cs) which gives B and C. Take AB in the compass (2C1), draw C (AB) (C1+Cs) which on the same side of BC as A. Op. (2R1+R2+4C1+2C3). S. 9;

places D on O (OA)
Draw AD (2R1+R2).
E. 6. 1 line, 2 circles.

V. On a given line as chord, to describe a segment of a circle containing a given angle LMN.

(a) Classical construction. Draw BD making with AB an angle equal to LMN; draw the perpendicular bisector of AB and the perpendicular at B to BD, these two perpendiculars intersecting in O; draw O (OA). Op. (6R,+3R2+11C,+8C3). S. 28; E. 17. 3 lines, 8 circles. In conducting the operations with economy we may reduce the symbol by (C1+C3).

(b) Geometrographic construction. Draw the circles A (AB), B (AB) (3C,+2C3), then L (AB) (C,+C3) which gives points K, M. Draw B (KM) cutting A (AB) in C, and C(KM) cutting same circle in D, (4C1+ 2C3). Draw the perpendicular bisector of AB (2R,+R2), and BD (2R1+R2) in · tersecting in E; finally draw E (EA) (2C+C3) which gives the segment required. Op. (4R,+2 R2+10C1+6C3). S. 22. E. 14. 2 lines, 6 circles.

2

VI. To draw a tangent to a circle of center O, at a point A on the circle. (a) The common construction is to erect a perpendicular to the radius at its extremity. Op. (6R,+3R2+C1+C3). S. 11; E. 7. 3 lines, 1 circle. (b) Geometrographic construction. any point of the given circle, draw B(BA) (2C1+C3) which cuts it again in C; draw A (AC) (2C1+C3) which cuts B (BA) in D. Draw DA the required tangent (2R,+R2).

B being

Op. (2R1+R2+4C,+2C3). S. 9; E. 6. 1 line, 2 circles.

B

VII. From a point A outside a circle of center O, to draw a tangent to the circle.

(a) Classical construction. Draw OA; describe the circumference which has OA for diameter and cuts the circumference in B and C, the

1

points of tangency. Draw AB and AC. Op. (8R,+4R2+4C1+3C3). S. 19; E. 12. 4 lines, 3

circles.

(b) Geometrographic construction. Draw M any diameter MN (R,+R2); draw M (OA), N(OA) (4C1+2C3) intersecting in P. Draw A (OP) (3C1+C3) which cuts the given circle in B and C, the points of tangency. Draw the tangents AB and AC. (4R1+2R2). Op. (5R1+3R2+7C1+3C3). E. 12. 3 lines, 3 circles.

S. 18;

N

VIII. To construct the mean proportional X between two given lines A and B. X2=A.B. Let A>B.

(a) (b) and (c) the three constructions commonly given; the first two are based on the properties of the segments of the hypotenuse of a right triangle made by the perpendicular from the vertex of the right angle, and the third is based on the property of the tangent to a circle and the segments of the secant drawn through the same external point as the tangent. They give for simplicity and exactness,

(a) S. 22; E. 14; (b) S. 28; E. 17; (c) S. 30; E.19. (d) Geometrographic construction. Draw any line (R2) and with any point of the line O as center, draw O(A), (2C1+C2+C3) which cuts the line in C and D. Draw C (B) (3C1+C1) which places E between C and D; draw E (B), (C1+C3) and through the intersections of these two circles draw the perpendicular bisector of CE cutting O (A) in F. CF is the mean proportional. Op. (2R1+2R2+6C1+C2+3C3) S. 14; E. 9. 2 lines, 3 circles.

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D

We can prove this by noticing that in the right triangle CFD, C F2=CL.CD=2.2A=A B.

B

IX. To divide a line AB in extreme and mean ratio.

(a) Classical construction. In making this construction as it is ordinarily done, we have Op. (6R,+3R2+11C,+9C3). S. 29; E. 17. 3 lines, 9 circles. By geometrographic principles it may be reduced to Op. (4R1+2R2+10C1+8C3). S. 24; E. 14. 2 lines, 8 circles.

There are a great number of constructions more simple than the classical construction, among them several geometrographic constructions. The following is one of them.

(b) Geometrographic construction. Draw A (AB) (2C,+C3), which

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cuts AB in D; draw D (AB) (C+Ca) cutting the first circle in F and C. Draw FC cutting AB in G. Draw G (AB) cutting FC in H. While the point of the compass is at G, take the length GB in the compass (C), then draw H (GB) (C+C3) cutting AB in M and M'. M is the point of internal division and M' the point of external division. Op. (2R,+R2+

6C1+4C3). S. 13; E. 8. S. 13; E. 8. 1 line, 4 circles. The simplification in this case is from 29 to 13.

That which precedes is sufficient to show the application of Geometrography. It is remarkable that it has been possible to simplify all the fundamental constructions, sometimes to a very great extent, as in the construction of a mean proportional from 28 to 14, and in the division of a line in extreme and mean ratio from 29 to 13.

The author, M. Lemoine, has applied geometrographic methods to a considerable number of other problems, and has also modified these constructions by allowing other instruments to be used, in particular the square or right triangle used by draughtsmen.

BIOLOGICAL SECTION

Conference of the Biological Section

Two sessions were held, Friday afternoon and Saturday afternoon, March 28 and 29, the latter being a joint session with the Michigan Academy of Science. Both sessions were well attended.

After calling the meeting to order and making announcements, the chairman, Mr. L. Murbach, made some remarks on the proposed amalgamation of the biological section with the Michigan Academy of Science. He said the biological section would better not surrender its identity by fusion with the Michigan Academy of Science, but that the Michigan Academy of Science might form a science-teaching section which the members of the biological section could join, leaving them the opportunity to hold their own conference at any time when the Michigan Academy of Science meets at a different time or place from the Schoolmasters' Club. He pointed out the advantages of the union as, membership in the larger body, opportunity to hear purely scientific papers, and the right to the publications of the proceedings. Pending the action of the Michigan Academy of Science, the Conference proceeded with its program, electing Dr. F. C. Newcombe, chairman, and Miss Genevieve Derby, secretary. Papers were presented as follows:

THE RELATION OF NATURE STUDY TO HIGH SCHOOL BIOLOGY

RAY A. RANDALL. ASSISTANT PRINCIPAL, GOShen (ind.) HIGH SCHOOL

Never before has the subject of science occupied the prominent place in which it stands now. Never has the progress of civilization been so rapid as the last century, during which time science has done so much for the world. Science is the mother of civilization. This advancement was brought about by the mental operations of observation, experiment, classification, deduction, and generalization.

Today all scientific training, all scientific knowledge must come through these primary conceptions and today these primary conceptions

must come through nature's door as in the past. For the child's nature, dependent upon his inherited impulses, necessitates the exercises of his powers through experiences similar to those which took part in the physical and mental development of his ancestors. The lifeless forms of Latin and Greek are no longer in the path of advancement. We now think in our own language, but the present century has yet to blot out the word or the form idea of the old regime. We think the child has the idea when he has only the form in which it is expressed. The present day idea of teaching the child new words by associating the word with the object I believe to be radically wrong in that it makes nature subjective, not objective. The object in the child's hand becomes a part of his experience and that experience expressed will bring the word desired.

Nature study furnishes a basis of reasoning, i. e. from particular to general, which applied to other studies makes real their notions. It puts the child in the right attitude for work, makes him independent in thought and action, and by its reactive influence moulds the character.

Upon recognizing the aim and importance of nature study in the grades we next turn our attention to the presentation of the subject, to the basis for work and the relation of the work to High School Biology. Many attempts to introduce nature study in the grades have failed, and I dare say it is due to a great extent to the method in vogue of leaving the science work to the grade teacher. To teach nature study in the right way a Col. Parker is needed, a person who can lead others to observe and experiment. Time and a great deal of it is needed. Time to locate the proper field of study, time to take the children to the fields, time to prepare and perform experiments, time to look over note-books with the individual. Systematic work is needed, not a heap of experiment without a definite aim in view. Material both for experiment and observation is need d and should be of the proper kind and plenty of it. In most Graded Schools at the present time nature work is taught by the teachers of the respective grades and in most cases, not knowing the subject they are instructed in the work by the superintendent or science teacher. Due to the nature of the case they cannot fulfill the requirement of a nature study teacher. A specialist born a naturalist would satisfy the condition, and since the importance of a right relation of the child to nature cannot be measured in dollars and cents, an argument against the expense due to the employing a special instructor for the town, has no basis. Again if we recognize nature study as being on a par with reading, language, history, etc., it demands a place in the curriculum and should be presented by a teacher whose acquaintance with the subject is as thorough as with other subjects. One instructor could handle from three to five grades giving six hours a week to a grade by devoting three periods of two hours each to each grade, with satisfaction.

In Biology which furnishes a large share of topics for nature work the natural whole may be the single animal or plant or several living objects taken together to form a society for a definite aim or purpose. Such units or wholes as above mentioned form a natural basis for deductive reasoning, comparisons and generalizations. The course of reasoning should be, in general, first, observation of life in the single thing and repeated recognition of the different fundamental laws: second, application of the laws to unfamiliar objects and life societies.

In the selection of topics strict scientific order should not control.

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