Bernhard Marks

“ An angle at the circumference is measured by half the arc on which it stands.
Let B A D be an angle whose vertex is in the circumference of the circle whose centre is C; then will it be measured by half the arc B D.
For through the centre draw the diameter A E, and join the points C and B.
The exterior angle E C B is equal to the sum of the angles B and B A C.
Because the sides C A, C B, are radii of the circle, they are equal, the triangle is isosceles, the angles B and B A C opposite the equal sides are equal, and the angle B A C is half of both. ”

“ If this view is correct, it is wrong to allow a pupil to reach the age of fourteen years without knowing even the alphabet of Geometry. He should be taught at least how to read it.
It certainly does seem probable, that if the youth who now leave school with so much Arithmetic, and no Geometry, were taught the first rudiments of the science, thousands of them would be led to the study of the higher mathematics in their mature years, by reason of those attractions of Geometry which Arithmetic does not possess. ”

“ Is an arc of a hundred and seventy-nine degrees greater or less than a semi-circumference?
Can there be more than three hundred and sixty degrees in a circumference?
If the circumference of circle 1 were divided into degrees, each degree would be so small an arc that it would look like a dot.
If a degree were divided into sixty equal parts, each part would be called a minute.
If a minute were divided into sixty equal parts, each part would be called a second.
How many degrees in the large circle of Fig. 2?
How many in the smaller one?
Has a large circle any more degrees than a small circle? ”

“ If the angle Red were smaller, and the angle Green larger, would the adjacent angles still be equal to two right angles?
Then,—
Any two adjacent angles are equal to two right angles.
If we draw the straight line c d where the edges of the square corners come together, what kind of angles will a c d, d c b, be? ”

“ The circumference of a circle is the curve which bounds it.
Name a straight line that joins two points in the circumference.
It is called a “chord.”
A chord is a straight line that joins two points of a circumference.
Read six chords in the diagram.
Which two of these chords pass through the centre?
They are called “diameters.”
A diameter is a chord that passes through the centre.
Name a line that joins the centre with a point of the circumference.
It is called a “radius.”— (Plural, radii.)
A radius is a straight line that joins the centre to a point of the circumference. ”

“ It is called an exterior angle.
Read three other exterior angles formed by the lines a b, c d, and e f.
Why is the angle c n f an exterior angle?
An exterior angle is one that lies outside of the parallels. ”

“ The prime object of school instruction is to place in the hands of the pupil the means of continuing his studies without aid after he leaves school. The man who is not a student of some part of God’s works cannot be said to live a rational life. ”

“ Children are taught to read, not for the sake of what is contained in their readers, but that they may be able to read all through life, and thereby fulfil one of the requirements of civilized society. So, enough of each branch of science should be taught to enable the pupil to pursue it after leaving school. ”

“ That the reason may be profitably addressed through the medium of Geometry at as early an age as seven years is asserted by no less an authority than President Hill of Harvard College, who says, in the preface to his admirable little Geometry, that a child seven years old may be taught Geometry more easily than one of fifteen. ”

“ An isosceles triangle has its two equal sides each three inches long, and its third side five inches long, what is its perimeter?
A right-angled isosceles triangle has its base five inches, and its hypothenuse seven inches long, what is its perimeter?
A square geography-cover is nine inches long on one side, how long all round? ”

“ It is easier for a child to comprehend what is than what is not.
Note C.—If the pupils are old enough, they may be shown that vertical lines cannot be parallel, but only seem so on account of their shortness and nearness to each other.
Note D.—This definition may be considered objectionable because rhomboid means like a rhomb. That the more general figure, the rhomboid, has been named from the more restricted one, the rhomb, is unfortunate, because it interferes with the symmetry of the nomenclature. ”

“ Why is m n g an acute angle?
What is an acute angle?
Here the attention is directed first to this particular angle; then this is taken as an example of its kind, and the idea generalized by describing the class. See also page 29.
Why are the lines e f and g h said to be parallel?
When are lines said to be parallel?
Many of the questions are intended to test the vividness of the pupil’s conception. See page 29.
Also page 78. If the circumference were divided into 360 equal parts, would each arc be large or small?
Many of the questions are intended to test the attention of the pupil. ”

“ It is certainly true that perception should receive a larger share of attention than the other faculties during the first school years; but it is equally certain that no faculty can be safely disregarded, even for a time. ”