quadgram

This is a table of type quadgram and their frequencies. Use it to search & browse the list to learn more about your study carrel.

quadgram	frequency
the sum of the	42
in the case of	32
is equal to the	26
the area of a	26
the area of the	24
it is well to	22
sum of the angles	22
on the other hand	21
for the purpose of	18
at the same time	17
the case of the	17
the leading propositions of	16
in the same way	16
the volume of a	16
seems to have been	16
the teaching of geometry	15
is one of the	15
the use of the	15
leading propositions of book	14
the mensuration of the	14
of the angles of	14
a straight line is	14
to the fact that	14
the nature of the	14
the fact that the	14
it is possible to	14
in connection with the	13
in the form of	13
area of a circle	13
in regard to the	13
the volume of the	13
by the use of	12
to the product of	12
by means of a	12
equal to the product	12
the construction of the	12
one of the most	12
it is easy to	12
at the close of	12
at the beginning of	12
it is interesting to	12
that a straight line	12
plane and solid geometry	11
angles of a triangle	11
it is evident that	11
by means of the	11
on the part of	11
equal to one another	11
the study of the	11
is said to have	11
the diameter of the	11
the length of the	11
the product of the	11
the product of its	11
it is probable that	10
is perpendicular to the	10
from the standpoint of	10
it is better to	10
the number of sides	9
the number of propositions	9
of the fact that	9
the square on the	9
as shown in the	9
the included angle of	9
straight line is the	9
of a square inch	9
of the one are	9
at the present time	9
in the same plane	9
so as to make	9
that the area of	9
the center of the	9
that there is no	9
the locus of a	9
included angle of the	9
to square the circle	9
the case of a	9
the size of the	8
the form of a	8
and the included angle	8
area of the circle	8
the value of pi	8
sides and the included	8
to a given line	8
of a triangle are	8
on the other side	8
the one are equal	8
to a class to	8
is from the greek	8
have been made to	8
are equal respectively to	8
the side of the	8
of its base by	8
the sides of a	8
by the help of	8
the other two sides	8
locus of a point	8
sides of a triangle	8
the angles of a	8
is greater than the	8
but it is not	8
the circumference of a	8
proposition relating to the	8
the edge of a	8
its base by its	8
triangles are congruent if	8
as a matter of	8
two sides and the	8
and this is the	8
two triangles are congruent	8
base by its altitude	8
in the use of	8
product of its base	8
the side of a	8
one are equal respectively	8
to have been the	8
the square root of	7
in such a way	7
it is true that	7
one of the best	7
to find the area	7
square on the hypotenuse	7
of a fourth dimension	7
angle of the other	7
angle of the one	7
proclus tells us that	7
proposition of plane geometry	7
in the way of	7
it is necessary to	7
what is meant by	7
of the teaching of	7
circumference of a circle	7
of a straight line	7
find the area of	7
mensuration of the circle	7
it will be found	7
the conservation of energy	7
to the sum of	7
in spite of the	7
of a triangle is	7
so far as to	7
the height of the	7
of a right triangle	7
attempts have been made	7
two sides of a	7
is seen in the	7
in a straight line	7
is from the latin	7
the value of the	7
the bottom of the	7
for the sake of	7
of the same kind	7
the use of a	6
from the fact that	6
the weight of the	6
the discovery of the	6
the foot of the	6
of the conservation of	6
are equal to one	6
a straight line and	6
from time to time	6
finding the area of	6
duplication of the cube	6
l b l c	6
the history of the	6
of the sides of	6
is to be proved	6
are congruent if two	6
we could find the	6
it is well known	6
attention to the fact	6
nair i z i	6
on one side of	6
interesting to a class	6
in terms of the	6
transmutation of the metals	6
two sides of the	6
this is one of	6
the first book of	6
such a way as	6
gothic designs employing circles	6
of the eighteenth century	6
proposition in plane geometry	6
at right angles to	6
the projection of a	6
of the nature of	6
a considerable number of	6
the powder of sympathy	6
is supposed to be	6
the elixir of life	6
a piece of paper	6
of the seventeenth century	6
the beginning of the	6
that it is the	6
equal to the sum	6
it would be possible	6
of algebra and geometry	6
to speak of the	6
if a straight line	6
of a circle is	6
foot of the perpendicular	6
it would not be	6
a large number of	6
in the fifth century	6
the circumference of the	6
for the reason that	6
a great deal of	6
designs employing circles and	6
measured by half the	6
the surface of a	6
line is the shortest	6
the locus of points	6
it should also be	6
as a center and	6
have been the first	6
the close of the	6
it will be noticed	6
it will be seen	6
the same circle or	6
over and over again	6
value of the ratio	6
a way as to	6
equal respectively to two	6
the case of two	6
come down to us	6
be found in the	5
than that of the	5
it is to be	5
in the united states	5
in the first place	5
two angles and the	5
should be noticed that	5
perpendicular to the plane	5
the trisection of an	5
been the first to	5
the part of the	5
that it was the	5
of an inch in	5
is known as the	5
a circle may be	5
in the same order	5
have the same ratio	5
in one of the	5
are to each other	5
be noticed that euclid	5
at the end of	5
to two right angles	5
the end of the	5
as far as possible	5
can easily be made	5
sum of the squares	5
to the area of	5
will be noticed that	5
of squaring the circle	5
to each other as	5
as in the case	5
the extremities of a	5
equal to two right	5
wrote a commentary on	5
propositions of plane geometry	5
on account of its	5
will be that of	5
tells us that the	5
of the nineteenth century	5
is an interesting exercise	5
the duplication of the	5
the surface of the	5
the mean proportional between	5
is well known that	5
the size of a	5
the plane of the	5
path between two points	5
it is not a	5
can be drawn to	5
the squares on the	5
cut by a transversal	5
half the sum of	5
the ratio of the	5
total for plane geometry	5
in connection with this	5
equal to half the	5
the solution of this	5
and the included side	5
in the hope that	5
trisection of an angle	5
the experience of the	5
teaching of elementary mathematics	5
to the nature of	5
circumference of the circle	5
construction of the regular	5
for the purposes of	5
would be possible to	5
the segments of the	5
of a regular inscribed	5
the transmutation of the	5
the construction of a	5
it should be noticed	5
perpendicular to a line	5
perpendicular to a plane	5
the best of the	5
a right angle is	5
the shortest path between	5
is the shortest path	5
extreme and mean ratio	5
of the sixteenth century	5
to a given circle	5
mean proportional between the	5
that it is impossible	5
that there is a	5
to the use of	5
in the middle ages	5
of the area of	5
and it is not	5
the lateral area of	5
angles and the included	5
and it is well	5
is measured by half	5
shortest path between two	5
will be found that	5
the teaching of mathematics	5
of the same size	5
it is helpful to	5
gives an account of	5
is that which has	5
of an isosceles triangle	5
the sum of two	5
and it will be	5
will be seen that	5
study of the subject	5
on the teaching of	4
between axiom and postulate	4
in the eighteenth century	4
or in equal circles	4
similar to the one	4
of geometry in the	4
of a regular polygon	4
to call attention to	4
triangle that has one	4
corresponding proposition in plane	4
circle or in equal	4
points of the compass	4
inscribed and circumscribed polygons	4
right angles are equal	4
of a given cube	4
a line perpendicular to	4
and the elixir of	4
is a straight line	4
perpendicular to the same	4
and one of the	4
a triangle that has	4
which have been made	4
of which the side	4
the circle and the	4
not seem to be	4
circle may be described	4
as the extremity of	4
in any of the	4
it has been suggested	4
with respect to the	4
in the same circle	4
from a to b	4
but we do not	4
angle of the triangle	4
the law of converse	4
same circle or in	4
this proposition is the	4
let us suppose that	4
of plane and solid	4
the study of geometry	4
the area of an	4
congruent if two sides	4
the proposition relating to	4
one side of a	4
conception of a fourth	4
the straightedge and compasses	4
these two propositions are	4
archimedes and his fulcrum	4
the angles of all	4
determine a straight line	4
of which it is	4
to be found in	4
is the mean proportional	4
shown in the engraving	4
in the british museum	4
a perpendicular to a	4
it is impossible to	4
and half an egg	4
is the limit of	4
than a right angle	4
it is difficult to	4
the same is true	4
of this proposition is	4
in equal circles equal	4
of the mensuration of	4
of a perpetual motion	4
a quarter of a	4
is equal to half	4
to the study of	4
the first of these	4
but at the same	4
angles a right angle	4
is that of the	4
of the other two	4
has been suggested that	4
and that it is	4
three sides of the	4
the circle is the	4
the motion of the	4
the hope that the	4
a given straight line	4
the meaning of the	4
is interesting to a	4
it is easily proved	4
that is to say	4
reduce the number of	4
the fixation of mercury	4
on the side of	4
volume of a given	4
the points of the	4
twice the volume of	4
seen in the case	4
the three sides of	4
has one of its	4
be drawn to a	4
a simple matter to	4
is the basis of	4
if two sides and	4
that which has its	4
law of the conservation	4
triangle is equal to	4
the position of the	4
the proposition about the	4
a few of the	4
a good deal of	4
a copy of the	4
half an egg more	4
it is said that	4
it is customary to	4
of the value of	4
a circle is a	4
with the study of	4
the corresponding proposition in	4
in the mensuration of	4
corresponding proposition of plane	4
that it is not	4
the properties of the	4
sides of a right	4
is that which is	4
the definition of a	4
of the basal propositions	4
one of two parallel	4
an account of a	4
by means of which	4
regular polygons of n	4
the fact that a	4
for the construction of	4
are cut by a	4
first book of euclid	4
in the works of	4
a commentary on euclid	4
area of a triangle	4
parallel to a given	4
be made to coincide	4
is perp to x	4
divided into three equal	4
the reductio ad absurdum	4
product of the segments	4
on a square inch	4
in any triangle the	4
the attention of the	4
the law of the	4
lateral area of a	4
from an external point	4
same is true of	4
side of a square	4
same time we must	4
it is claimed that	4
the centers of the	4
this phase of the	4
attention called to the	4
tells us that he	4
straight line and a	4
the one about the	4
a point equidistant from	4
the same ratio as	4
all right angles are	4
is a right triangle	4
quadrature of the circle	4
be equal to the	4
it is found that	4
the diameter of a	4
invented a perpetual motion	4
volume of a sphere	4
a quadrant of the	4
of the diameter of	4
in the hands of	4
the included side of	4
a matter of fact	4
angle of a triangle	4
included side of the	4
may be found in	4
be seen that the	4
each other as the	4
on the surface of	4
illustration illustration illustration illustration	4
of the circle is	4
lines in the same	4
experience of the world	4
a regular inscribed polygon	4
when we come to	4
distant from the center	4
of the first folio	4
center and any given	4
respectively to two sides	4
a class to have	4
equal to a given	4
at a very early	4
is the case in	4
the bible and testament	4
two straight lines are	4
angles of a polygon	4
have come down to	4
that geometry is not	4
center of the circle	4
the edge of the	4
that has one of	4
the extremity of a	4
the proofs of the	4
one of its angles	4
lines are cut by	4
to become a millionaire	4
its angles a right	4
triangle that which has	4
the same base and	4
one of the equal	4
four for the third	4
of a sphere is	4
of its angles a	4
straight angles are equal	4
the intersection of two	4
the same time we	4
in the fact that	4
said to have been	4
brought into contact with	4
of the locus of	4
an inch in diameter	4
greater than the third	4
i have endeavored to	3
has the sanction of	3
been known to the	3
been suggested that the	3
that can easily be	3
of the middle ages	3
a basis for the	3
but the proof is	3
he is said to	3
the sense of touch	3
the value of geometry	3
applications of this proposition	3
a preliminary to the	3
is shown in the	3
what are not the	3
the language of the	3
let it swing about	3
in his mathematical recreations	3
may be considered as	3
than two right angles	3
as de morgan says	3
the equilateral triangle and	3
some of the most	3
as that of the	3
bottom of the crucible	3
of the school grounds	3
in this connection that	3
of a century ago	3
distinction between axiom and	3
of the american mathematical	3
to about one hundred	3
that pythagoras discovered the	3
course in solid geometry	3
the efforts of the	3
do not attempt to	3
to find the side	3
length of the circumference	3
to let fall a	3
the fact that there	3
the limit of a	3
one cent for the	3
the bisector of an	3
the definition is not	3
the measure of the	3
line perpendicular to a	3
cut off the corners	3
as it is generally	3
and a few of	3
euclid does not use	3
line is a line	3
a solution of the	3
of the leading propositions	3
the purpose of increasing	3
this is the first	3
line as a radius	3
reasons for studying geometry	3
for the teacher to	3
from the study of	3
angle formed by two	3
a little more than	3
pyramid is equal to	3
quarter of a dollar	3
points determine a straight	3
should be given to	3
than is the case	3
on the other two	3
by a straight line	3
it is related that	3
with any given point	3
in the same direction	3
in a bowl of	3
the compasses and ruler	3
find the center of	3
the faces of the	3
are unequal in the	3
questions like the following	3
in the schools of	3
the two triangles are	3
that there are two	3
for the measuring of	3
number of sides of	3
that all right angles	3
angle of a regular	3
two angles of a	3
as a preliminary to	3
be divided into three	3
is perpendicular to a	3
to one of the	3
in the american high	3
same base and the	3
a straight line not	3
is easily shown that	3
is a right angle	3
paper so as to	3
if two parallel lines	3
so that it may	3
all straight angles are	3
more than is necessary	3
an ordinary paper protractor	3
it is a good	3
are not to be	3
the computation of the	3
of two parallel planes	3
as shown by the	3
be that of the	3
angles of all the	3
the seven follies of	3
of the triangle and	3
a diameter bisects the	3
terms of the radius	3
the number of edges	3
the same thing are	3
exterior angles of a	3
natural history of hell	3
that can be drawn	3
a triangle are concurrent	3
by half the sum	3
that some of the	3
given in the textbook	3
be found that the	3
needless to say that	3
the same straight line	3
with most of the	3
on the opposite side	3
are congruent if the	3
means of which he	3
which has its three	3
it is not difficult	3
to the end of	3
in area to a	3
propositions in plane geometry	3
it should always be	3
tenths of an inch	3
equals are added to	3
the corresponding proposition of	3
a hole in the	3
off the corners of	3
a regular polygon of	3
that it should be	3
an angle is the	3
a given point in	3
it is advisable to	3
that which does not	3
the figure is a	3
the analogous proposition of	3
do not seem to	3
are some of the	3
it is not worth	3
may be described with	3
from a fixed point	3
the powers of the	3
to run a line	3
to two sides and	3
a triangle is equal	3
side of the other	3
a certain number of	3
which have the same	3
diameter of a circle	3
there are not wanting	3
the proof of the	3
is exactly in line	3
if it were not	3
beyond the powers of	3
of the subject in	3
if i can prove	3
universal medicine and the	3
the opening of the	3
and of course the	3
the first century b	3
the natural history of	3
equal in area to	3
measurement will show that	3
sense of the word	3
given distance from a	3
the american high school	3
one side of the	3
no apology for giving	3
be described with any	3
is one third of	3
to any required length	3
will be equal to	3
not in the same	3
the second part of	3
parallel to an element	3
so as to have	3
about one hundred thirty	3
same as that of	3
we may think of	3
i have no doubt	3
side of the one	3
as to the nature	3
of the old and	3
the fifth century b	3
of a new sense	3
nor the study of	3
it is needless to	3
the radius of a	3
as the number of	3
the search for a	3
the hands of a	3
to bear in mind	3
area of a trapezoid	3
of a similar kind	3
is equal to a	3
that it may be	3
of the incommensurable case	3
give only the ability	3
of this kind of	3
it may have been	3
the corner at c	3
area of the surface	3
that the number of	3
and to give a	3
line is perpendicular to	3
hole in the hand	3
in contact with the	3
a considerable amount of	3
i can prove this	3
on the island of	3
of the segments of	3
a triangle whose base	3
is equal to two	3
is said to be	3
th of an inch	3
shown in this figure	3
through a given point	3
the alkahest or universal	3
attempts to square the	3
of a class in	3
protractor may be used	3
to the work in	3
the diameter is given	3
opposite the equal sides	3
of two straight lines	3
form part of the	3
square on a line	3
of the circle and	3
a sphere may be	3
in the study of	3
diameter of the circle	3
this is done by	3
bible might be written	3
mathematical recreations and problems	3
if we could find	3
to the same line	3
any given point as	3
in the direction ab	3
of alexander the great	3
in the same ratio	3
thousandth of an inch	3
have twice the volume	3
angle by taking the	3
it is perpendicular to	3
of the point as	3
in the seventeenth century	3
lies in the fact	3
and any given line	3
were known to the	3
it may be said	3
drawn from a point	3
injected into the boiler	3
a question of population	3
perpendicular bisectors of the	3
one of the sides	3
angle is equal to	3
a proposition relating to	3
an important part in	3
perpendicular to the line	3
to inscribe a regular	3
any given line as	3
the number of proved	3
the purposes of elementary	3
given line as a	3
can be drawn through	3
and the construction of	3
the method of analysis	3
from two given points	3
it is doubtful if	3
before the christian era	3
was supposed to be	3
a class in geometry	3
the same number of	3
of the compasses and	3
it was not until	3
the object of this	3
that a diameter bisects	3
how to become a	3
purposes of elementary geometry	3
at a greater distance	3
the top of the	3
it might be difficult	3
of the definitions of	3
diameter of the earth	3
one of the many	3
fathom the depths of	3
through a hole in	3
at a given distance	3
but most of them	3
so it is with	3
of the first century	3
on a straight line	3
straightedge and the compasses	3
use of the compasses	3
the first is the	3
algebra and geometry are	3
in reality it is	3
conduct of a class	3
in the state of	3
the needs of the	3
therefore the sum of	3
a plane parallel to	3
is the shortest line	3
the line of centers	3
from the educational standpoint	3
the fact that if	3
the nature of these	3
in the high school	3
would do well to	3
the spirit of the	3
this is not the	3
the comprehension of the	3
the definitions of geometry	3
of the subject is	3
equal respectively to the	3
the same as in	3
as to make the	3
out of all the	3
the sphere and cylinder	3
given point as a	3
the teaching of elementary	3
is by no means	3
seven wise men of	3
form a conception of	3
perpendicular to the base	3
diameter and the circumference	3
in the midst of	3
the areas of two	3
we should have a	3
the band of sponge	3
the same as that	3
and some of the	3
three times the diameter	3
to a given angle	3
two lines drawn from	3
if l b l	3
so as to lie	3
taking the ratio of	3
of a circle to	3
connection with the study	3
as well as the	3
perpendicular to the axis	3
the straight line is	3
that geometry is a	3
the writings of the	3
an angle of a	3
base and the same	3
just as we may	3
i can prove that	3
described with any given	3
is parallel to the	3
one can fathom the	3
to the pythagorean theorem	3
then l b l	3
to the mensuration of	3
to form a conception	3
ratio of the circumference	3
the first folio shakespeare	3
of the same base	3
of all the faces	3
same ratio as their	3
for the number of	3
as one of the	3
of each of these	3
a small budget of	3
in this and the	3
if we cut through	3
interest to the subject	3
the diagonal of a	3
the seven wise men	3
can be made to	3
be the number of	3
diagonal of a square	3
and in this way	3
of the inscribed and	3
re et praxi geometrica	3
employing circles and bisected	3
be noticed that the	3
by taking the ratio	3
the great french philosopher	3
round the corner at	3
the essential features of	3
from a given external	3
we can find the	3
as it is called	3
to those who have	3
to the corresponding proposition	3
the propositions of geometry	3
explain what is meant	3
it will not be	3
the second chapter of	3
it may also be	3
in the teaching of	3
the number of applications	3
can fathom the depths	3
called to the fact	3
call attention to the	3
translation of the greek	3
the relation of the	3
than c without violating	3
the perimeter of a	3
to find the center	3
lines containing the angle	3
be used as a	3
information in regard to	3
equal the sum of	3
construction of a perpetual	3
the help of the	3
perpendicular to the other	3
in this chapter to	3
a rule for finding	3
of the intercepted arcs	3
angles have the same	3
of the five regular	3
perpendicular to one of	3
no royal road to	3
a special case of	3
possibility of a new	3
that have been made	3
the invention of printing	3
squares on the other	3
an angle equal to	3
surface of the earth	3
perpendicular to a given	3
from the point of	3
figure is a parallelogram	3
an account of the	3
the scale of the	3
greater than that of	3
parallel lines are cut	3
to that of a	3
the angle between the	3
analogous proposition of plane	3
line from a given	3
find the side of	3
are subtracted from equals	3
centers of the faces	3
a circle is the	3
a point in the	3
is also true of	3
should give only the	3
as seen in the	3
the length of a	3
the section is a	3
number of simple exercises	3
the perpendicular bisectors of	3
that the volume of	3
perpendicular to the edge	3
in favor of the	3
to a given straight	3
a square which shall	3
is easily proved that	3
in all such cases	3
the reason for this	3
the drawing of a	3
a point in a	3
the height of a	3
of the most important	3
in the plane of	3
it is not so	3
be seen from the	3
fact that there is	3
side of a regular	3
the idea of the	3
that the pupil has	3
the action of the	3
may be made to	3
the accumulation of water	3
in his perpetuum mobile	3
of the regular polygons	3
we are justified in	3
there can be no	3
search for a perpetual	3
that the machine will	3
by a plane parallel	3
square which shall be	3
why geometry is studied	3
are shown in the	3
two parallel lines are	3
the base of the	3
it seems to have	3
and to show that	3
seven follies of science	3
the diameter and the	3
all the straight lines	3
from a point in	3
only the ability to	3
equal to the radius	3
a few years ago	3
less than a right	3
known to the ancients	3
into three equal parts	3
a given external point	3
in the same parallels	3
this can be done	3
times on a square	3
the five regular polyhedrons	3
as to lie on	3
if two straight lines	3
the statement that a	3
in order to have	3
arc of a circle	3
plane perpendicular to the	3
with which we are	3
proposition asserts that if	3
this kind of work	3
space of three dimensions	3
of the theory of	3
be made to move	3
exactly in line with	3
the square and circle	3
extremity of a line	3
the center of similitude	3
height of the tree	3
in the number of	3
be equal to one	3
may be inscribed in	3
to show that the	3
two points determine a	3
but there is a	3
in extreme and mean	3
the corners of the	3
to the pupil as	3
diameter bisects the circle	3
a class to see	3
common tangents to two	3
surface of a sphere	3
for a class to	3
it was thought that	3
the thirteen books of	3
it is easier to	3
the way in which	3
a certain amount of	3
the perimeters of the	3
of the way in	3
to say that the	3
the quadrant used for	3
part of the work	3
inscribed in a semicircle	3
the limit of the	3
of one per cent	3
is the locus of	3
a good plan to	3
two right triangles are	3
cos o x r	3
the science of the	3
if it should give	3
speak of the locus	3
point as a center	3
distance from a fixed	3
the axioms and postulates	3
than the sum of	3
used as a compass	3
interior angles are equal	3
if equals are subtracted	3
an arc of a	3
in the drawing of	3
this book is written	3
history of greek mathematics	3
the arrangement of the	3
the textbook in geometry	3
is good educational policy	3
the third side of	3
is now in the	3
in which it is	3
a good reason for	3
might be difficult to	3
one third of a	3
right angles to the	3
the meaning of a	3
we come to consider	3
de re et praxi	3
may be seen in	3
rectilinear figures book ii	3
by the pythagorean theorem	3
in the sense of	3
large part of the	3
of the squares on	3
at the opening of	3
it is easily shown	3
propositions of solid geometry	3
of the regular pentagon	3
the other side of	3
half the product of	3
of the most interesting	3
length of the diameter	3
an angle formed by	3
equals are subtracted from	3
well to ask a	3
how to find the	3
the influence of the	3
if equals are added	3
of the surface of	3
cent for the first	3
an acute angle is	3
of the present day	3
to the needs of	3
it would seem that	3
from a given point	3
propositions of book iii	3
of a polygon of	3
a plane surface is	3
the american mathematical society	3
from the foot of	3
that they should be	3
it would have been	3
it may be stated	3
a given distance from	3
area to a given	3
the perpendicular bisector of	3
circles and bisected angles	3
a plane through the	3
the beginning of this	3
and the same altitude	3
third side of the	3
of the equal sides	3
the two straight lines	3
lines drawn from a	3
the machine will not	3
it must not be	3
the state of new	3
of any of the	3
the quadrature of the	3
by a broken line	3
an angle of the	3
a center and any	3
alkahest or universal solvent	3
it is certain that	3
state of new york	3
is thus described by	3
no one can fathom	3
from any point on	3
let fall a perpendicular	3
so as to form	3
purpose of increasing the	3
in speaking of the	3
than the third side	3
we shall now consider	3
in addition to the	3
a circle of which	3
two intersecting straight lines	3
a large part of	3
by the name of	3
is also of interest	2
of a quarter of	2
easily followed by the	2
a pair of compasses	2
in heaven and earth	2
who may care to	2
which have been offered	2
that we are justified	2
through a pipe of	2
a reasonable amount of	2
this is not a	2
hypatia who is the	2
that might better be	2
shadow may be made	2
the truth of the	2
and the circumference of	2
on the floor of	2
they have a few	2
the proposition was known	2
fifteen years of age	2
to the mind of	2
the great basal propositions	2
as stated in the	2
straight lines and circles	2
may of course be	2
pythagoras fled to megapontum	2
equal to that of	2
a subject in which	2
all those who have	2
a finite number of	2
referring to the figure	2
of course be understood	2
line that can be	2
he should be asked	2
three times as long	2
may be said that	2
problem may be given	2
so that they shall	2
to the exterior angle	2
be borne in mind	2
price of the horse	2
if the watch is	2
a right triangle is	2
more distant from the	2
in the art of	2
two books of euclid	2
parallel ruler of the	2
relation of algebra to	2
of the thing defined	2
contained by the straight	2
propositions of book vii	2
occupied the attention of	2
this is an interesting	2
for attempting to change	2
as the circumference and	2
number of equal parts	2
standpoint of strict logic	2
discussed in this volume	2
to the extracting of	2
practically little or no	2
the society for the	2
as has already been	2
and is said to	2
it is interesting and	2
by david eugene smith	2
of the proposition about	2
line and a circle	2
of course no one	2
want of a better	2
by n points in	2
to the hypotenuse and	2
the whole history of	2
the inhabitants of flatland	2
more concrete than a	2
becomes zero and the	2
or the locus of	2
well to call attention	2
by the fact that	2
o is called the	2
that the locus of	2
be effected by the	2
of the work and	2
side of the first	2
equals in the same	2
of the picture a	2
triangle is that which	2
is how many times	2
in some of the	2
for the effecting of	2
of the straightedge and	2
the circle book iv	2
the picture a man	2
is the case with	2
what is known as	2
merely a matter of	2
the circumference cut off	2
the results of similar	2
borne in mind that	2
a straight line set	2
is open to the	2
us that pythagoras discovered	2
is equivalent to a	2
and think of the	2
floor of the schoolroom	2
if we have two	2
it appears that the	2
from the center are	2
the opposite sides of	2
in a country where	2
regular polygon of which	2
by the number of	2
at the top of	2
that the square on	2
illusions of the senses	2
these will now be	2
was known to the	2
a few knitting needles	2
the sake of geometry	2
of interest in the	2
to calculate the length	2
pupils in the american	2
in writing of the	2
would not be a	2
that the bible and	2
the great majority of	2
better understood than the	2
there is a certain	2
are not in the	2
many parts of the	2
may be seen from	2
equally distant from the	2
as to what are	2
of a universal medicine	2
before the time of	2
und methode des planimetrischen	2
and then of s	2
the middle of the	2
of the few propositions	2
is shown in this	2
ratio and proportion book	2
of the same material	2
diameter of the circumscribed	2
if we say that	2
results true of that	2
seen through a hole	2
in the thirteenth century	2
for the average pupil	2
in higher mathematics it	2
of the plans which	2
of the patent office	2
attempts which have been	2
straight line is a	2
and there is no	2
between algebra and geometry	2
to use a book	2
parallel is drawn through	2
base has the same	2
walk along ab until	2
by half the intercepted	2
the basis of selection	2
the discovery of this	2
the exterior angle becomes	2
has the same length	2
this can conveniently be	2
of the last century	2
is added to the	2
case was that of	2
to be classed with	2
it should be said	2
of the cube as	2
volume of a cylinder	2
the principles of the	2
going so far as	2
terms of the three	2
area of an isosceles	2
it were not for	2
there is no more	2
is too difficult for	2
to point the hour	2
a small wooden pin	2
the royal society of	2
tells us that pythagoras	2
our solar system is	2
point equidistant from the	2
in order to fix	2
that this book is	2
placed a stake at	2
a line is drawn	2
motion might be obtained	2
been made to explain	2
it is quite possible	2
may be apparently enlarged	2
when we attempt to	2
of a point in	2
pleasing to the eye	2
of which had been	2
the attempts which have	2
will cut off the	2
a occupies the same	2
phase of the subject	2
if we are to	2
in elementary geometry the	2
to make geometry more	2
a point b is	2
was heron of alexandria	2
be applied to the	2
are many variations of	2
there is a little	2
know that the idea	2
are equal and parallel	2
a plane which is	2
it is sufficient to	2
of the oblique prism	2
a pupil has a	2
it was used for	2
through the center of	2
fourth term of the	2
one line can be	2
that the straight line	2
angle is an angle	2
a polygon of n	2
the introduction of the	2
first is greater than	2
he gave a rule	2
to ask a class	2
at the vertices of	2
the circle has been	2
are perpendicular to each	2
and to select the	2
be apparently enlarged by	2
make sure that the	2
a circle to a	2
a consideration of the	2
beginning of solid geometry	2
relating to plane and	2
the volume of any	2
the story goes that	2
wise men of greece	2
the circumference and whose	2
to attack the exercises	2
this proposition to thales	2
of the first is	2
as we shall see	2
to mean both the	2
is to be considered	2
but as to the	2
the proposition is not	2
that the result is	2
a boy or girl	2
should be recommended to	2
the discovery of new	2
and at the same	2
sum of the sides	2
a man of acknowledged	2
number of proved propositions	2
as to make a	2
of the principle of	2
it must be confessed	2
the pupil will be	2
as the limit of	2
has been made in	2
of the present will	2
of equal base and	2
angle equal to a	2
sum of the three	2
of these forms are	2
insisted upon is that	2
that under some conditions	2
falling upon one wheel	2
is now proposed to	2
the subject has been	2
will of course be	2
three hundred years ago	2
of the tree is	2
for the basal propositions	2
right angle is called	2
and b cannot be	2
case of two circles	2
at its extremity is	2
the date of these	2
use of the triangle	2
realize what life in	2
plan of drawing a	2
so that it is	2
definite point in the	2
approximating the value of	2
plane parallel to the	2
was supposed to have	2
of which are equal	2
a little interest to	2
a straight line as	2
that in order to	2
is doubtful if the	2
the end in view	2
elements on one side	2
the author of the	2
that is in the	2
is the fact that	2
side of the subject	2
und die sechs planimetrischen	2
the school of plato	2
polygon of which the	2
of the one equals	2
a side of the	2
vertical angle of the	2
improved upon this by	2
is sometimes called the	2
had a strange fascination	2
in some other way	2
looked upon in geometry	2
mean the bounding line	2
identically equal in the	2
the ratio of similitude	2
congruent if the hypotenuse	2
simple as easily to	2
relating to the exterior	2
it is so easy	2
to do with the	2
from the definition of	2
the whole is greater	2
plane of the two	2
is that relating to	2
are said to be	2
and the projection of	2
have an angle of	2
is called a right	2
one of them is	2
extremities of a given	2
is called the center	2
the radius in question	2
are far beyond the	2
those who wish to	2
line perpendicular to another	2
b and d are	2
and circumscribed polygons as	2
gave a rule for	2
teachers have any such	2
the angles at the	2
characters so fine that	2
close of book iv	2
propositions de morgan selected	2
if from a point	2
to fall into a	2
hypotenuse and an adjacent	2
as the products of	2
fall outside the circle	2
millionth of an inch	2
the exterior angles of	2
adapted to the needs	2
its way into elementary	2
the island of sicily	2
the study of a	2
the success of the	2
two cents for the	2
of the proposition is	2
the important thing is	2
the reason is that	2
part of the pupil	2
have the same extremities	2
is interesting to note	2
angle is called a	2
should be called to	2
from the latin translation	2
discovery of the liquefaction	2
the eye to the	2
the proof of this	2
not good policy to	2
a cube that should	2
would not be sufficient	2
in the course of	2
much more than this	2
a line as a	2
twentieth globe above us	2
the line joining the	2
any one of them	2
half an ounce of	2
study of the regular	2
circumference of any circle	2
it is a fact	2
as a special case	2
precisely the same length	2
should be as few	2
present value of the	2
there is also a	2
the point from which	2
or the same as	2
following are some of	2
an enormous amount of	2
partly that of the	2
and an adjacent angle	2
the secondary schools of	2
the method by which	2
this is a very	2
it is very obvious	2
of most of the	2
which the pupil is	2
necessary for a beginner	2
that he discovered the	2
is to be observed	2
needs of the beginner	2
drawn through two given	2
of double the number	2
about the line of	2
they should be so	2
in a cistern of	2
of two sides of	2
subject in the curriculum	2
france and de paolis	2
selected with their corollaries	2
who does not know	2
the claim that the	2
a pair of these	2
is also perpendicular to	2
finite number of square	2
his budget of paradoxes	2
of the work of	2
is necessary for the	2
the teacher may have	2
is well that we	2
out of the large	2
to one of two	2
and b as centers	2
true of that world	2
the following are some	2
so that it has	2
be asked of a	2
it has been fully	2
where there is a	2
the elasticity of the	2
of the reductio ad	2
a straight line from	2
same thing may be	2
on this subject are	2
the tree is found	2
and in the course	2
of the propositions that	2
it is always interesting	2
are subtended by equal	2
to have a class	2
on the two sides	2
and it shows that	2
of the ratio of	2
with one of the	2
received so much attention	2
the name of the	2
effecting of a perpetual	2
area of a parallelogram	2
in the same class	2
written at rate of	2
and the second is	2
be postponed until after	2
familiar to the pupil	2
is the use of	2
applications of these propositions	2
can be done by	2
area of the school	2
a sphere is equal	2
begin to curve at	2
a straight line to	2
as we have seen	2
between the hour and	2
as a basis for	2
the propositions of euclid	2
even in regard to	2
base of an isosceles	2
problem is impossible by	2
the latter part of	2
in so far as	2
obtain results true of	2
describe a square which	2
and a pi r	2
visualize a solid from	2
proofs of the basal	2
ratio of diameter to	2
sin o y r	2
has this to say	2
gallons of liquid air	2
an excellent illustration of	2
of the sense of	2
as accurate as the	2
straight line on a	2
are found in the	2
better suited to the	2
as has been suggested	2
this proposition is not	2
of approximating the value	2
were at one time	2
the greek word for	2
valve in the center	2
it at right angles	2
way in which he	2
the actual cost and	2
the regular hexagon is	2
proposition to be proved	2
surface of the ice	2
be in the same	2
the chapters and the	2
would be at once	2
which has no part	2
there is also another	2
is not of much	2
the floor of the	2
area of a regular	2
the homogeneity of space	2
what is the height	2
of in your philosophy	2
that should have twice	2
essential features of a	2
very little more than	2
he might have postulated	2
ourselves in the midst	2
took hold of the	2
with a velocity of	2
quadrant of the sixteenth	2
the lay of the	2
in the school course	2
significance of the negative	2
plane there can be	2
on page shows how	2
more than a year	2
and it is interesting	2
the product of their	2
the shadow may be	2
these forms are shown	2
on the score of	2
that geometry was taught	2
required to construct an	2
so far as they	2
his account of his	2
that are necessary to	2
of gold or silver	2
the second and third	2
in the performance of	2
hand side of the	2
the hypotenuse and an	2
straight line in space	2
to curve at p	2
picture a man is	2
side of the tube	2
shows how it was	2
is not so simple	2
is well to state	2
be called upon to	2
between two lines that	2
to be observed that	2
of the required cube	2
man lift himself by	2
twice the size of	2
in terms of a	2
one of the old	2
the length of xy	2
make no apology for	2
then let it swing	2
in this country to	2
small wooden pin at	2
true of the man	2
and they will be	2
has to do with	2
the early history of	2
a second angle of	2
the reasons for studying	2
when the diameter is	2
be described in terms	2
less than two right	2
before the eighteenth century	2
one line parallel to	2
may have been the	2
central angles are measured	2
one perpendicular to a	2
be followed by a	2
to be proved in	2
there have been numerous	2
by one line meeting	2
of the volume of	2
have invented a perpetual	2
of compasses and straightedge	2
seen in a method	2
adjacent angle of the	2
on the other must	2
this reduces to the	2
shall be less than	2
the semicircle is the	2
the hypotenuse must equal	2
in his account of	2
from going very far	2
we are not quite	2
the first part of	2
with the compasses and	2
the same idea in	2
but in spite of	2
is said by proclus	2
many times her own	2
father of the hypatia	2
in the line of	2
now before me a	2
illustration another interesting application	2
to divide a line	2
the greater angle is	2
for both of these	2
we derive pleasure from	2
any other subject in	2
to grasp the meaning	2
of the three angles	2
of the dark ages	2
the pressure would at	2
perpendicular to that line	2
is quite possible that	2
locus of points equidistant	2
very nature of the	2
medicine and the elixir	2
although some of the	2
like the following may	2
i and its propositions	2
too abstract to be	2
stage in his progress	2
is needless to say	2
measured by their intercepted	2
the definition for the	2
could square the circle	2
to say how it	2
it is equal to	2
the principle involved in	2
circumference cut off by	2
on page shows a	2
some of the questions	2
connection with this proposition	2
angle sine cosine tangent	2
the vertices of the	2
is better to have	2
must have had a	2
was one of the	2
so simple as easily	2
in the year the	2
taught mathematics in the	2
curve through some higher	2
and do not lie	2
we have left the	2
mensuration of the prism	2
to construct an arc	2
his problems with his	2
the hypotenuse of a	2
a crystal formed by	2
and the printing press	2
in france and de	2
line is drawn through	2
their corollaries as necessary	2
the work relates chiefly	2
there are many more	2
used for the same	2
of pure and applied	2
let abc be the	2
it was because of	2
case they would be	2
each parallel to a	2
of a point equidistant	2
the past few years	2
and in the same	2
part of the eighteenth	2
he did not know	2
the following is a	2
that there must be	2
some of the weights	2
ball tells us that	2
the value of all	2
the principle of the	2
would there not be	2
at the rate of	2
more things in heaven	2
until a point b	2
the school should give	2
ropes out of sea	2
gives the area of	2
the view from the	2
geometry that we have	2
how to attack the	2
the whole is equal	2
which he claimed to	2
efforts at improving euclid	2
of the propositions in	2
say that we have	2
iron rods are hinged	2
of a quadrilateral are	2
cut the circle twice	2
way to introduce this	2
the effecting of a	2
it will of course	2
of the semicircle is	2
hypotenuse must equal the	2
the introduction to geometry	2
seen in the illustration	2
the circumference to the	2
the diameter will weigh	2
be drawn through two	2
straight lines containing the	2
is subtended by the	2
the great french mathematician	2
the proposition that the	2
an illustration of the	2
if two angles and	2
construct an equilateral triangle	2
the effect of all	2
the unaided eye can	2
line of work that	2
the manner of a	2
the use of these	2
of their numerical values	2
squares on any two	2
compute the length of	2
and whose altitude is	2
corollaries as necessary for	2
of a point as	2
of the circumscribed circle	2
the number of basal	2
as necessary for a	2
demands another type of	2
of the numerical measures	2
circumference of the segment	2
one of the planes	2
a very early day	2
by means of this	2
meet in a point	2
now in the british	2
us suppose that the	2
measured on a line	2
to the subject to	2
hold of the wheel	2
and in every case	2
there is a very	2
as few as possible	2
the distance across a	2
of the size of	2
point in the plane	2
as the quotient of	2
question as to why	2
attempts were made to	2
himself by the straps	2
the frustum becomes a	2
the basal propositions of	2
to any practical mechanic	2
is not difficult to	2
exterior angle of a	2
the straight lines containing	2
is true that the	2
until the invention of	2
the division of a	2
in the theory of	2
not the reasons for	2
the twentieth globe above	2
it mean that a	2
sum of the areas	2
is bound to produce	2
position of the sun	2
or it may be	2
that our treatment of	2
to the action of	2
of the number of	2
how it was used	2
until you reach a	2
pupils in the high	2
a conception of a	2
how many times as	2
for the various angles	2
earth and good will	2
be answered by the	2
by the diameter and	2
after the discovery of	2
fell in love with	2
is given by proclus	2
on the scale of	2
of diameter to circumference	2
thence again descend on	2
moved about in space	2
and the number of	2
how it can be	2
column of de luc	2
but they hung a	2
that just as we	2
if p is at	2
of the tube with	2
the sums are equal	2
the royal asiatic society	2
to the measuring of	2
the subject is in	2
what the problem is	2
is an account of	2
to move in the	2
what would be the	2
and in a short	2
is a simple matter	2
in america it is	2
a tangent to a	2
euclid und die sechs	2
of the faces of	2
to have been merely	2
an angle less than	2
pupils in manual training	2
put a stake at	2
planes are perpendicular to	2
of this problem is	2
the line as a	2
it is not of	2
need not be discussed	2
are needed in the	2
it is very certain	2
interest of their pupils	2
the drawing of figures	2
passes through the mid	2
been done by the	2
to have given the	2
by professor de morgan	2
get the figure of	2
the straight line and	2
we are not so	2
which is helpful to	2
in the earliest times	2
the progress of science	2
to produce this effect	2
the beginning of geometry	2
is a special case	2
the rest may be	2
we are told that	2
order to have the	2
of the cylinder is	2
a correspondent of the	2
of two congruent triangles	2
wish to run a	2
semicircle is a right	2
there would have been	2
of precisely the same	2
fourth dimension or the	2
the study of such	2
the real purpose of	2
and of the most	2
following is an interesting	2
of which we have	2
not quite sure of	2
be moved about in	2
other two sides of	2
discovered the construction of	2
of it as a	2
the figure of a	2
at p and q	2
full appreciation of the	2
of a conic surface	2
angle between the hour	2
it is proper to	2
the same way fix	2
given in connection with	2
his mathematical recreations and	2
reason for studying geometry	2
will thus be seen	2
sides of a quadrilateral	2
when viewed under a	2
offered and which he	2
security for the last	2
to the amount of	2
is here that the	2
which the side is	2
the fact that all	2
that two points determine	2
pi r the volume	2
writing of the same	2
the proof is not	2
has been left as	2
reduction in the number	2
if two planes are	2
to be within the	2
measure of the circle	2
in the secondary schools	2
thing may be said	2
the division of the	2
its bases multiplied by	2
as to make such	2
in any other branch	2
this proposition is practically	2
that of finding the	2
that of the other	2
is perpendicular to one	2
with the case of	2
by the simple process	2
that the sum of	2
is meant by ratio	2
the sides about the	2
or shall it be	2
to be the case	2
a study of the	2
is not used as	2
to the second fractional	2
by doubling the number	2
by the force of	2
there is also some	2
general determined by n	2
that it shall be	2
of those who have	2
show the relation of	2
been caused by the	2
exactly what we have	2
means of a very	2
let it be required	2
formed by replacing each	2
the points of division	2
shortest distance between two	2
and in view of	2
it is not necessary	2
the point may even	2
taught in the schools	2
the definition of line	2
does it mean that	2
received into the other	2
that we may use	2
and the volume is	2
tangents to two circles	2
any triangle the greater	2
who lived in the	2
the next she sold	2
as to why geometry	2
that have come down	2
the describing of circles	2
the projection of the	2
and a side of	2
of a segment is	2
are liable to be	2
measure the length of	2
also given by proclus	2
here again we may	2
which case they would	2
other as the squares	2
postponed until after the	2
to measure to the	2
little or no friction	2
they placed a stake	2
number of propositions in	2
pipe of double the	2
that has preceded it	2
through p parallel to	2
be based upon the	2
at the foot of	2
straight lines in the	2
the first century a	2
it is with geometry	2
by the perimeter of	2
would not be difficult	2
be introduced to the	2
primarily for the sake	2
things in heaven and	2
of a box is	2
is that we should	2
us that it was	2
to the next valve	2
is that of a	2
propositions have been discovered	2
triangle in terms of	2
ignorant both of the	2
to draw a line	2
instead of the wound	2
do this by the	2
that the pupil can	2
the first is greater	2
a subject that has	2
impossible by elementary geometry	2
geometry is not taught	2
that all straight angles	2
to the comprehension of	2
but it is evident	2
of the side of	2
of sides of the	2
a velocity of about	2
line is that which	2
in the center is	2
we are allowed to	2
time to time to	2
and that this was	2
and why do we	2
they can be made	2
two planes perpendicular to	2
of a regular pyramid	2
above the level of	2
planes is perpendicular to	2
both of the nature	2
the hypatia who is	2
it may be a	2
is that of archimedes	2
and solid geometry is	2
and of the results	2
we must make the	2
error does not exceed	2
is it better to	2
that the teacher may	2
we have just given	2
of a cone of	2
in characters so fine	2
little more than three	2
then at the figure	2
all the work for	2
about the year b	2
made much of the	2
before the british association	2
the exterior angle of	2
as the bounding line	2
special case of the	2
from which bc makes	2
it is generally called	2
five colors of marble	2
refused to allow a	2
six books of euclid	2
when in reality it	2
is evident that the	2
lead them to a	2
which is then at	2
and the circumference cut	2
further from the centre	2
it enables us to	2
that may lead to	2
of the most common	2
it is only on	2
and the sum of	2
about the sum of	2
difficult to define as	2
is a curious fact	2
in the next illustration	2
considered the nature of	2
good will toward men	2
of the world is	2
the knowledge of the	2
to construct an equilateral	2
of the famous sir	2
if for no other	2
the conclusion that their	2
times as long as	2
it from one point	2
is sufficient to point	2
the greatest of the	2
the column of water	2
the bisecting of angles	2
to visualize a solid	2
nature of the problem	2
earth only one inch	2
all things are possible	2
is at least as	2
backward on the sun	2
bottom of the vessel	2
were told that the	2
such a procedure is	2
library of george a	2
is not ready for	2
on the road to	2
through some higher space	2
in the sixteenth century	2
in even the finest	2
angles equal to two	2
in a right triangle	2
class to have attention	2
nature of the exercises	2
perpendicular can be drawn	2
morgan gives the following	2
models in solid geometry	2
that have been suggested	2
call the attention of	2
in this case the	2
of the one equal	2
at this point in	2
due to the pythagoreans	2
author of the natural	2
lift himself by the	2
cylinder is equal to	2
the conduct of a	2
value of the above	2
approximate value of pi	2
equal to the area	2
side of the vertex	2
a logical sequence of	2
cuts it at right	2
geometry is not studied	2
a line through p	2
for teachers who may	2
to enable us to	2
seem to the pupil	2
history of the teaching	2
the conclusion that the	2
he wrote the first	2
us to square the	2
the mental uplift that	2
if to this we	2
to move backward on	2
by their intercepted arcs	2
add a little interest	2
inscribe a regular polygon	2
is both equilateral and	2
in such a simple	2
altitude is equal to	2
to prove that a	2
if the hypotenuse and	2
the number of times	2
proclus calls attention to	2
right angle is a	2
is so easy to	2
the nail problem a	2
propositions of book iv	2
how do we know	2
royal academy of sciences	2
which have come down	2
the illustration given on	2
by means of straight	2
with the preceding one	2
has already been discussed	2
as difficult to define	2
was the first to	2
simplifies the treatment of	2
is equivalent to the	2
all the faces is	2
on any two corresponding	2
run a line through	2
quarter of a century	2
the thing that is	2
a straight line makes	2
thing are equal to	2
it is advantageous to	2
to the corollary that	2
as far as the	2
a manuscript of the	2
a triangle are equal	2
is thought to be	2
of the circumscribed cylinder	2
by the french mathematician	2
straps of his boots	2
by looking at the	2
we may have cross	2
n points in space	2
the daughters of men	2
the approximate value of	2
it should be noted	2
the sphere and the	2
by the circumference of	2
find the value of	2
and by means of	2
the number of human	2
upon the subject are	2
from the sense of	2
the father of the	2
square the circle or	2
constructing a cube that	2
which have been already	2
then of s sides	2
it might be well	2
books ii and v	2
perpetual motion it is	2
come to consider the	2
of mathematics in the	2
and one of his	2
and good will toward	2
b is perp to	2
is less than degrees	2
is tangent to the	2
by calculating the perimeters	2
of the sides being	2
for the case of	2
of the circle as	2
as to give a	2
a number of these	2
that has come down	2
determined by n planes	2
so that the angle	2
he claimed to be	2
have in my possession	2
is the heroine of	2
the special case of	2
to make the subject	2
was by means of	2
seem to be the	2
defined either as a	2
solution of the problem	2
that it might be	2
had studied in egypt	2
actual cost and present	2
of boys and girls	2
congruent if two angles	2
the fusion of all	2
down to us from	2
may be used as	2
sides of the right	2
is very certain that	2
like a piece of	2
may be said for	2
doubling the number of	2
on the sphere and	2
angled triangle that which	2
circumscribed polygons as the	2
being conscious of it	2
in the center of	2
definition of straight line	2
calculate the side of	2
of the twelfth century	2
occupies the same space	2
part of the proof	2
respectively to two angles	2
in terms that are	2
segments from the foot	2
the circle as a	2
discovery of this proposition	2
things are possible to	2
do more than mention	2
we are forced to	2
that it affords a	2
to be understood by	2
regular inscribed and circumscribed	2
the diagonals of a	2
in a few seconds	2
angles at the base	2
it is good educational	2
made by a plane	2
one of the three	2
in general determined by	2
of a regular hexagon	2
volume of the sphere	2
about in space without	2
a spherical triangle is	2
be seen in the	2
tells us that a	2
put on the board	2
cylinder is pi r	2
the very nature of	2
the radius of the	2
is not suited to	2
whole is greater than	2
to make use of	2
such a proof is	2
the same idea is	2
opening of the tube	2
to erect a perpendicular	2
to have been a	2
may be used in	2
any other branch of	2
this is the reason	2
the same side of	2
easier to grasp than	2
it is only a	2
is always interesting to	2
of the powder of	2
reduces to the pythagorean	2
the direction of the	2
a line such that	2
to be taught as	2
so simple that it	2
the following is the	2
economy of time and	2
it is in the	2
the latin translation of	2
could find the area	2
to which we are	2
to have attention called	2
so fine that the	2
base is equal to	2
of models in solid	2
to two angles and	2
replacing each edge of	2
the finding of the	2
in regard to what	2
something to do with	2
circle is measured by	2
value of pi exactly	2
a line parallel to	2
the standpoint of strict	2
the same as the	2
equal to each other	2
which was supposed to	2
invisible to the naked	2
this was done by	2
there is to be	2
times as much as	2
on the hypotenuse must	2
the question of teaching	2
cost and present value	2
from the greek polys	2
without breadth or thickness	2
seems to have come	2
subtended by the greater	2
present population of the	2
perpendicular to the floor	2
the band and chain	2
by word of mouth	2
royal road to geometry	2
would have been no	2
the close of book	2
all the spirit of	2
spoke of it as	2
which is the locus	2
propositions usually given in	2
right triangles are congruent	2
y x tan o	2
it is the basis	2
of an angle of	2
the building up of	2
the rest of the	2
in which case they	2
if there is an	2
may not be as	2
the world in which	2
and it would not	2
that there can be	2
the conditions under which	2
is easy to measure	2