trigram

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

trigram	frequency
one of the	94
a straight line	83
the area of	62
the number of	57
the sum of	54
in the same	53
the fact that	51
it is not	50
by means of	48
the case of	47
but it is	47
is equal to	45
the use of	45
and it is	45
of the circle	44
sum of the	42
equal to the	42
on the other	40
of a circle	39
the study of	39
area of a	36
so as to	36
some of the	36
that it is	35
part of the	35
it is well	35
it will be	34
in the case	34
of a triangle	34
of the other	34
of the subject	34
it may be	32
the volume of	32
of the same	31
perpendicular to the	30
the product of	29
that of the	29
side of the	28
in regard to	28
to have been	28
there is no	27
in other words	27
of the most	27
so that the	27
to find the	27
it is a	26
tells us that	26
the teaching of	26
the value of	25
as to the	25
area of the	25
of the angles	25
this is the	25
is to be	24
of all the	24
the nature of	24
that there is	24
seems to have	23
in which the	23
the length of	23
there is a	23
the construction of	23
it should be	23
the center of	23
of the triangle	23
in connection with	22
in plane geometry	22
of an inch	22
is well to	22
of the first	22
it is the	21
the other hand	21
sides of a	21
of this kind	21
relating to the	21
angle of the	21
of a square	20
for the purpose	20
is as follows	20
have been made	20
at this time	20
a perpetual motion	20
the mensuration of	20
the side of	20
to a given	20
at the same	20
of plane geometry	20
a right angle	20
and this is	20
case of the	19
the purpose of	19
algebra and geometry	19
a piece of	19
is perpendicular to	18
and so on	18
straight line is	18
it is possible	18
of a line	18
the diameter of	18
which have been	18
in order to	17
of the one	17
the same time	17
would not be	17
is said to	17
the leading propositions	17
perpendicular to a	17
the circumference of	17
shown in the	17
fact that the	17
two straight lines	17
the same way	17
it would be	17
is one of	17
to a class	17
study of the	17
the beginning of	17
the size of	17
nature of the	17
angles are equal	17
propositions of book	16
teaching of geometry	16
the proof is	16
it is also	16
this is a	16
from the center	16
seen in the	16
leading propositions of	16
this proposition is	16
volume of a	16
sides of the	16
the locus of	16
it has been	16
two right angles	16
to say that	16
is from the	16
said to have	16
angles of a	16
use of the	16
the form of	16
a right triangle	16
a given line	15
as well as	15
so far as	15
of a regular	15
of the two	15
and in the	15
to make a	15
in elementary geometry	15
of the cube	15
is greater than	15
that geometry is	15
the angles of	15
of this proposition	15
and that the	15
the pythagorean theorem	15
it is interesting	15
mensuration of the	15
to each other	15
a square inch	14
as here shown	14
is possible to	14
to the fact	14
at the beginning	14
it is better	14
the basis of	14
the surface of	14
and the other	14
it is easy	14
so that it	14
of elementary geometry	14
most of the	14
in the form	14
the sides of	14
equal to one	14
which it is	14
attention to the	14
in such a	14
an angle of	14
that we have	14
to the same	14
side of a	14
connection with the	13
volume of the	13
this is not	13
is that of	13
is not so	13
construction of the	13
of the proposition	13
the method of	13
regard to the	13
to make the	13
of the earth	13
have been the	13
be noticed that	13
the proposition is	13
of solid geometry	13
the present time	13
those who have	13
as that of	13
the idea of	13
the discovery of	13
from the greek	13
the close of	13
as in the	13
two sides of	13
supposed to be	13
a circle is	13
and the included	13
to one another	13
a matter of	13
of the best	13
it might be	13
of the sides	13
on account of	13
the definition of	13
to the pupil	13
is not a	13
have the same	13
value of the	13
a class to	13
equal to a	13
that it was	13
to the product	12
the edge of	12
of a point	12
of the present	12
the eighteenth century	12
at the close	12
out of the	12
in this connection	12
by the use	12
length of the	12
triangles are congruent	12
of which the	12
diameter of the	12
at the present	12
as shown in	12
means of a	12
plane and solid	12
it is to	12
of the world	12
what is the	12
a number of	12
is interesting to	12
all of the	12
the sense of	12
and of the	12
would have been	12
the meaning of	12
the basal propositions	12
the straight line	12
of the propositions	12
may have been	12
it is probable	12
that a straight	12
the ratio of	12
that which is	12
to have the	12
is easy to	12
of the work	12
in this way	12
the name of	12
is in the	12
is seen in	12
it is evident	12
to the other	12
the other two	12
the part of	11
in the school	11
means of the	11
number of sides	11
is that which	11
known to the	11
we do not	11
product of the	11
to the base	11
the other side	11
to the one	11
a pair of	11
to use the	11
on the part	11
to show that	11
and solid geometry	11
the theory of	11
in the first	11
has not been	11
that which has	11
in a plane	11
is the shortest	11
that of a	11
value of pi	11
in which it	11
can easily be	11
in spite of	11
to be proved	11
there are many	11
squaring the circle	11
the square on	11
as to make	11
case of a	11
is evident that	11
of the regular	11
as it is	11
square the circle	11
the same as	11
is found in	11
as already stated	11
that the area	11
be found in	11
from a point	11
any of the	11
product of its	11
is an interesting	11
number of propositions	11
due to the	11
it is easily	10
of the old	10
the proof of	10
a triangle is	10
on the side	10
angle of a	10
of a straight	10
the middle ages	10
a series of	10
the included angle	10
an isosceles triangle	10
one of these	10
of the class	10
is better to	10
and the same	10
the result is	10
the work of	10
greater than the	10
in this case	10
that he has	10
be able to	10
square on the	10
a and b	10
many of the	10
that he had	10
idea of the	10
at right angles	10
that the pupil	10
geometry in the	10
two triangles are	10
that has been	10
at one time	10
at that time	10
the circle and	10
given in the	10
of an angle	10
can be drawn	10
as far as	10
from the standpoint	10
in terms of	10
to speak of	10
that they are	10
a kind of	10
the standpoint of	10
in one of	10
a fourth dimension	10
will be found	10
the united states	10
that we may	10
of the various	10
is probable that	10
place in the	10
powder of sympathy	10
the height of	10
parallel to the	10
straight lines are	10
geometry is not	10
to be a	10
the attention of	10
application of the	10
will be the	10
the first is	10
which may be	10
the time of	10
the same plane	10
a perpendicular to	10
the same thing	10
of the second	10
is the case	9
be made to	9
speak of the	9
the amount of	9
of the following	9
by saying that	9
center of the	9
straight line and	9
found in the	9
in a circle	9
some of them	9
be possible to	9
to give a	9
made by the	9
the projection of	9
the results of	9
of a perpetual	9
of such a	9
and with a	9
the nineteenth century	9
are congruent if	9
the end of	9
from the latin	9
known as the	9
if equals are	9
may be used	9
the solution of	9
at this point	9
a great deal	9
a b c	9
the point of	9
the whole bible	9
point in the	9
with respect to	9
the following are	9
equal respectively to	9
to square the	9
in the way	9
the square root	9
included angle of	9
he does not	9
the history of	9
one side of	9
of the fact	9
a given point	9
in the second	9
it must be	9
the circle is	9
is a very	9
find the area	9
discovery of the	9
other two sides	9
the first to	9
proposition relating to	9
it as a	9
in which he	9
of its base	9
of a right	9
it is necessary	9
and it will	9
is given by	9
should also be	9
of the original	9
than that of	9
that we are	9
to have a	9
the subject is	9
we have a	9
of the diameter	9
are equal to	9
edge of a	9
it is usually	9
to do this	9
will not be	9
of a few	9
in solid geometry	9
circumference of the	9
line is the	9
seems to be	9
they may be	9
the one are	9
which is the	9
locus of a	9
large number of	9
is well known	9
and one of	9
to prove that	9
elixir of life	9
history of the	9
circumference of a	9
by the teacher	9
the seventeenth century	9
to the subject	9
on one side	9
end of the	9
the experience of	9
the science of	9
the principle of	8
to consider the	8
the fourth dimension	8
is necessary to	8
has been suggested	8
proclus tells us	8
if it were	8
by its altitude	8
of the circumference	8
the reason for	8
is shown in	8
what is meant	8
size of the	8
surface of the	8
the equilateral triangle	8
in the plane	8
and when we	8
are equal respectively	8
its base by	8
in the middle	8
by the help	8
is a circle	8
is that the	8
the way of	8
there can be	8
to measure the	8
the same ratio	8
if it is	8
it does not	8
form of a	8
one of them	8
the following is	8
in this country	8
be seen that	8
the most important	8
between two points	8
of the greek	8
of the teaching	8
given by the	8
as a radius	8
a little later	8
may be made	8
of perpetual motion	8
from a to	8
the idea that	8
it is more	8
the squares on	8
circle is the	8
of geometry as	8
is the same	8
to prove the	8
of those who	8
has been made	8
as a matter	8
the existence of	8
in this direction	8
the effect of	8
and it was	8
similar to the	8
and in this	8
to the plane	8
considerable number of	8
is supposed to	8
sides and the	8
the properties of	8
it is now	8
of the nature	8
applications of this	8
one are equal	8
and that it	8
is less than	8
reductio ad absurdum	8
illustration of the	8
been made to	8
of the angle	8
that have been	8
it is found	8
base by its	8
in the use	8
on a line	8
to a plane	8
it is very	8
is known as	8
parallel to a	8
to the end	8
two sides and	8
on the subject	8
of the ratio	8
a line is	8
of the three	8
ratio of the	8
much of the	8
a triangle are	8
is the one	8
these two propositions	8
say that a	8
the foot of	8
of the definitions	8
the law of	8
may be given	8
the knowledge of	8
a regular polygon	8
to be the	8
account of the	8
we wish to	8
to a line	8
the definitions of	8
it would not	8
conservation of energy	8
the problem is	8
be found that	8
that may be	8
of a polygon	8
there have been	8
the proofs of	8
instead of the	8
in the following	8
the help of	8
of the teacher	8
the weight of	8
the drawing of	8
seem to be	8
the propositions of	8
the relation of	8
as a center	7
a special case	7
may be a	7
is a right	7
in a few	7
of the area	7
for the reason	7
may also be	7
the figure is	7
on the whole	7
a plumb line	7
more than a	7
of the kind	7
point in a	7
not in the	7
to the area	7
attempts have been	7
the measuring of	7
to those who	7
in the fifth	7
from a given	7
we know that	7
definition of a	7
the work is	7
to get the	7
called to the	7
to become a	7
there are several	7
the limit of	7
that there are	7
of these two	7
form of the	7
a few of	7
we should have	7
right angles to	7
distance from a	7
there is also	7
the same circle	7
the case in	7
view of the	7
the same kind	7
line as a	7
they would be	7
the teacher to	7
a point in	7
the diameter and	7
as to be	7
the bottom of	7
a fact that	7
spite of the	7
the areas of	7
budget of paradoxes	7
that in the	7
bottom of the	7
it is true	7
and at the	7
the straight lines	7
this kind of	7
it is only	7
the first book	7
is parallel to	7
proof of the	7
an account of	7
height of the	7
these are the	7
of the perpendicular	7
nair i z	7
the fifth century	7
that such a	7
that he could	7
it was not	7
the plane of	7
little more than	7
should not be	7
the extremities of	7
a pi r	7
in higher mathematics	7
we may have	7
is true that	7
use of a	7
follies of science	7
the points of	7
the teacher of	7
bisulphide of carbon	7
it was the	7
edge of the	7
the corresponding proposition	7
of finding the	7
of two lines	7
of geometry in	7
is meant by	7
such a way	7
that they were	7
which had been	7
there is an	7
but there is	7
of the cylinder	7
on the hypotenuse	7
applied to the	7
transmutation of the	7
so that they	7
was made by	7
it seems to	7
is called the	7
the high school	7
perhaps the most	7
given on page	7
far as to	7
square root of	7
the result of	7
from that of	7
of the sphere	7
the third side	7
of the plane	7
in a straight	7
proposition of plane	7
in the center	7
the first of	7
this is an	7
the sake of	7
the equal angles	7
of a sphere	7
may be considered	7
the problem of	7
the line of	7
the applications of	7
i can prove	7
it cannot be	7
a quarter of	7
we are not	7
the conservation of	7
this may be	7
the hands of	7
be equal to	7
is helpful to	7
he might have	7
of a plane	7
we could find	7
to attempt to	7
not seem to	7
to the sum	7
such as is	7
two thousand years	7
the same size	7
inscribed in a	7
of a fourth	7
in this respect	7
in speaking of	7
two parallel lines	7
is not the	7
of which is	7
for the sake	7
of the pupil	7
the sixteenth century	7
and the circumference	7
for finding the	7
equivalent to the	7
each of the	7
shall it be	7
to see the	7
is called a	7
the pupil has	7
a quantity of	7
is not of	6
of this problem	6
can be made	6
are equal and	6
the transmutation of	6
employing circles and	6
a circle may	6
of two parallel	6
the purposes of	6
weight of the	6
because it is	6
the needs of	6
thought to be	6
a plane surface	6
for this purpose	6
to the line	6
to take the	6
based upon the	6
a list of	6
it is said	6
a given circle	6
is a good	6
of algebra and	6
such a plan	6
of an isosceles	6
one of its	6
reduces to the	6
of one of	6
and a half	6
this has been	6
it is generally	6
the works of	6
be that of	6
one of his	6
of the ancients	6
way as to	6
as much as	6
is easily proved	6
at first sight	6
attention called to	6
measured by half	6
he did not	6
same circle or	6
it should also	6
because of the	6
teaching of elementary	6
metals into gold	6
the elixir of	6
the art of	6
equidistant from the	6
circle may be	6
the best of	6
great deal of	6
teaching of mathematics	6
of the surface	6
number of the	6
a class in	6
could find the	6
in the usual	6
designs employing circles	6
the human mind	6
the lateral area	6
equal to half	6
case of two	6
so we may	6
illustration illustration illustration	6
it can be	6
if two sides	6
a way as	6
for plane geometry	6
followed by a	6
the preceding proposition	6
it is therefore	6
the hope that	6
it will not	6
by a transversal	6
will be noticed	6
if he had	6
down to us	6
over and over	6
the great french	6
part of this	6
duplication of the	6
of various kinds	6
to the present	6
it from the	6
and over again	6
the vertical angle	6
the possibility of	6
said to be	6
applications of the	6
conception of a	6
that the teacher	6
have been suggested	6
that we should	6
point of view	6
how many times	6
to a point	6
when it is	6
lines in the	6
propositions of plane	6
and the figure	6
account of a	6
of the squares	6
of the science	6
a large number	6
class in geometry	6
they are not	6
of the problem	6
that it may	6
angle is the	6
the influence of	6
and only one	6
by half the	6
the first century	6
in the direction	6
axioms and postulates	6
the books of	6
this was done	6
right angle is	6
and if the	6
textbook in geometry	6
to construct an	6
of the base	6
upon which the	6
which we have	6
to one of	6
proposition in plane	6
and a few	6
drawn from the	6
that if two	6
foot of the	6
the equal sides	6
from the fact	6
interesting to a	6
of a class	6
euclid did not	6
in this figure	6
projection of a	6
it as the	6
may be said	6
of the metals	6
opening of the	6
gothic designs employing	6
treatment of proportion	6
mean proportional between	6
of the wheel	6
terms of the	6
that can be	6
the question of	6
the force of	6
and in a	6
at any rate	6
at the end	6
to show the	6
the compasses and	6
of elementary mathematics	6
the word is	6
the powder of	6
of a given	6
close of the	6
of the last	6
surface of a	6
congruent if two	6
the author of	6
the spirit of	6
in equal circles	6
and we have	6
of the sixteenth	6
but they are	6
the opening of	6
been the first	6
time to time	6
he would have	6
there may be	6
propositions that are	6
piece of paper	6
i z i	6
each of these	6
if we could	6
of new york	6
cut by a	6
the three sides	6
they should be	6
for studying geometry	6
of the school	6
it would have	6
in favor of	6
one of two	6
it is helpful	6
which we are	6
it is impossible	6
distance from the	6
of the elements	6
of the word	6
of the eighteenth	6
line perpendicular to	6
if a straight	6
the reason that	6
a center and	6
of the seventeenth	6
to the work	6
ratio and proportion	6
must have been	6
in his mathematical	6
from time to	6
of the basal	6
this is one	6
the trisection of	6
would be possible	6
a few years	6
on the same	6
an acute angle	6
and we should	6
the perimeter of	6
we say that	6
the straightedge and	6
in line with	6
may be seen	6
locus of points	6
first book of	6
the figure of	6
are said to	6
there are two	6
will be seen	6
that it will	6
in this chapter	6
parts of the	6
the other end	6
as a result	6
a considerable number	6
the extremity of	6
of the conservation	6
b l c	6
the same base	6
beginning of the	6
would be the	6
the language of	6
most of them	6
this is also	6
the proposition that	6
by a plane	6
that he was	6
l b l	6
respectively to two	6
as early as	6
the exterior angle	6
to be found	6
between the two	6
finding the area	6
equal to two	6
must not be	6
the intersection of	6
development of the	6
first century a	6
this and the	6
be given to	6
in view of	6
of the great	6
in the year	6
come down to	6
for the same	6
an interesting exercise	6
to do with	6
may be so	5
is equivalent to	5
it had been	5
easily be made	5
plane of the	5
the work in	5
is difficult to	5
to make geometry	5
to the edge	5
this phase of	5
the progress of	5
determine a plane	5
and which is	5
the pupil will	5
a triangle that	5
solution of the	5
of liquid air	5
x and y	5
a little more	5
of this theorem	5
the proposition about	5
proportional between the	5
a subject that	5
geometry of the	5
the most famous	5
book of euclid	5
of a large	5
of which it	5
in geometry in	5
regular polygons of	5
is of no	5
and that this	5
will be that	5
be difficult to	5
size of a	5
the statement that	5
to the next	5
one that is	5
that all the	5
of time and	5
contained by the	5
for the purposes	5
a simple matter	5
are to each	5
made by a	5
within a circle	5
the success of	5
which does not	5
fact that a	5
and by the	5
should always be	5
the difference between	5
in the united	5
to take a	5
if we have	5
found that the	5
a modification of	5
if we are	5
was supposed to	5
an inch in	5
half an egg	5
properties of the	5
three sides of	5
wrote a commentary	5
angles and the	5
straight angles are	5
or to the	5
the hypotenuse and	5
in these words	5
that this was	5
two given points	5
value of geometry	5
the whole of	5
of a century	5
the subject in	5
account of its	5
lines drawn from	5
position of the	5
asserts that if	5
the definition is	5
of similar triangles	5
and if we	5
that we must	5
and the second	5
contact with the	5
for it is	5
has been the	5
in any triangle	5
of the theorem	5
mathematics in the	5
than the third	5
is only a	5
to half the	5
is perp to	5
solution of this	5
the object of	5
of these propositions	5
interest to the	5
they do not	5
the circle in	5
is capable of	5
each other as	5
way in which	5
in order that	5
he should be	5
of a spherical	5
first folio shakespeare	5
of the prism	5
may be stated	5
the line ab	5
make use of	5
section of the	5
there must be	5
and some of	5
a line as	5
to the corresponding	5
lies in the	5
be drawn to	5
from any point	5
the diagonal of	5
in this work	5
be obtained from	5
the top of	5
construction of a	5
the right angle	5
powers of the	5
for a few	5
us that the	5
suggested that the	5
the results are	5
lines perpendicular to	5
that if the	5
enable us to	5
the thing defined	5
that there was	5
the action of	5
solid geometry is	5
for the third	5
is a little	5
at the bottom	5
the radius of	5
geometry is a	5
the schools of	5
some of these	5
far as possible	5
to the use	5
formed by the	5
quarter of a	5
of this nature	5
and mean ratio	5
regular polygon of	5
angle formed by	5
the story of	5
there are also	5
special case of	5
noticed that euclid	5
proved to be	5
the sphere and	5
a commentary on	5
number of times	5
to the whole	5
to do so	5
the opposite side	5
proved by the	5
the truth of	5
and when the	5
of the simplest	5
three times the	5
the british museum	5
and the point	5
when we have	5
less than the	5
to decimal places	5
proposition is not	5
of s sides	5
the faces of	5
he was not	5
the regular hexagon	5
is due to	5
of course be	5
from the ground	5
of the frustum	5
of the greatest	5
of it as	5
and it would	5
of the pupils	5
inch in diameter	5
upon the subject	5
intersection of two	5
this does not	5
reasons for studying	5
of the radius	5
as a line	5
has been a	5
one hundred thirty	5
to prove it	5
of the line	5
are as follows	5
are perpendicular to	5
is easier to	5
in which case	5
should be noticed	5
less than a	5
in the great	5
center and any	5
it is quite	5
should be made	5
c pi r	5
illustrations of this	5
the segments of	5
in the american	5
is thus described	5
to which the	5
law of converse	5
inscribed and circumscribed	5
to be taken	5
to call attention	5
of a wheel	5
lateral area of	5
squares on the	5
in the figure	5
the included side	5
the cause of	5
question as to	5
measure the distance	5
the most interesting	5
the same length	5
if we know	5
a pupil in	5
books of euclid	5
of the right	5
in any of	5
attention of the	5
the same altitude	5
the duplication of	5
the right triangle	5
proposition may be	5
a new sense	5
we have the	5
of the term	5
propositions of geometry	5
perpetual motion is	5
is true of	5
to draw a	5
upon this subject	5
trisection of an	5
by doubling the	5
to the second	5
is also a	5
describe a circle	5
the same order	5
of the required	5
the first great	5
for the teacher	5
in the hope	5
adapted to the	5
and that he	5
the five regular	5
are cut by	5
be less than	5
terms that are	5
can prove that	5
extreme and mean	5
the methods of	5
such as the	5
worth while to	5
proposition is the	5
the upper base	5
point as a	5
form of proof	5
an equilateral triangle	5
of the compass	5
run a line	5
it is an	5
of the square	5
is measured by	5
an angle is	5
but this is	5
treatment of the	5
in three dimensions	5
have had a	5
the school grounds	5
to two right	5
sum of two	5
the shortest path	5
l a l	5
the application of	5
of the segments	5
a polygon of	5
segments of the	5
the mean proportional	5
of the equal	5
quantity of the	5
part of it	5
total for plane	5
is probably the	5
to construct a	5
as seen in	5
connection with this	5
well known that	5
of a new	5
the pupil is	5
that is to	5
to believe that	5
gold or silver	5
to the class	5
by taking the	5
sir kenelm digby	5
i have no	5
the first place	5
a century ago	5
the division of	5
a circle to	5
bible and testament	5
the most valuable	5
any number of	5
the present day	5
of a certain	5
meaning of the	5
enables us to	5
of the nineteenth	5
that it might	5
that he would	5
divided into three	5
path between two	5
two or three	5
p and q	5
is the figure	5
through the center	5
to introduce the	5
the bounding line	5
the circle as	5
of a rectangle	5
that it has	5
used as a	5
in the engraving	5
a b and	5
one third of	5
a regular inscribed	5
as they are	5
called by the	5
professor de morgan	5
all that is	5
of the circumscribed	5
at a given	5
and there is	5
extremity of a	5
if two angles	5
be seen in	5
to that of	5
but as a	5
to be learned	5
of the ancient	5
written on the	5
that if we	5
they will be	5
polygons of n	5
too difficult for	5
wrote the first	5
to a circle	5
may not be	5
to the pythagoreans	5
but it will	5
the position of	5
done in the	5
regular inscribed polygon	5
shortest path between	5
say that the	5
even the most	5
although it is	5
been suggested that	5
if he is	5
of its angles	5
that the machine	5
a fixed point	5
is much more	5
will find that	5
more than one	5
of any kind	5
ratio accepted by	5
to compute the	5
call attention to	5
first century b	5
heron of alexandria	5
may be found	5
of course this	5
angles equal to	5
the subject of	5
fixation of mercury	5
half the sum	5
gives the following	5
given by proclus	5
are to be	5
with the radius	5
the first folio	5
that the circle	5
for a beginner	5
law of the	5
to define a	5
is a parallelogram	5
point of the	5
experience of the	5
speaking of the	5
a given angle	5
plane perpendicular to	5
to the nature	5
drawn to a	5
given straight line	5
of squaring the	5
for the construction	5
to prove this	5
the old and	5
best of the	5
by the marquis	5
all of these	5
of a cylinder	5
and the angle	5
study of geometry	5
as to have	5
proof of this	5
a stake at	5
from the point	5
in book i	5
extremities of a	5
and they are	5
in the next	5
according to the	5
two angles and	5
point to a	5
at the top	5
translation of the	5
gives an account	5
if we take	5
a question of	5
the base of	5
by the greeks	5
that some of	5
in some cases	4
which is now	4
of which he	4
instead of a	4
on page shows	4
be considered as	4
in the eighteenth	4
if this is	4
it shows that	4
a quadrant of	4
to be true	4
faces of a	4
a rule for	4
a method of	4
the remainders are	4
is the basis	4
in this proposition	4
the search for	4
then it is	4
it has not	4
as a postulate	4
the same is	4
the pupil who	4
of the third	4
as de morgan	4
a subject of	4
that the word	4
if we cut	4
has one of	4
that it shall	4
is a line	4
the postulate that	4
a very simple	4
explanation of the	4
of the preceding	4
we come to	4
on the surface	4
first of these	4
that a triangle	4
in the other	4
are proportional to	4
that the thing	4
which the side	4
but at the	4
an external point	4
for which the	4
neither of which	4
in a textbook	4
if we consider	4
triangle is equal	4
of the simple	4
to divide a	4
one who is	4
circle or in	4
is liable to	4
to be more	4
the man who	4
as a solid	4
to solid geometry	4
expressed in modern	4
none of these	4
as a practical	4
the textbook in	4
pass through the	4
the geometry of	4
of all these	4
in laying out	4
was known to	4
corresponding proposition of	4
being in the	4
in a given	4
suppose that the	4
in this and	4
considerable amount of	4
effort is made	4
of plane and	4
for the second	4
a plane through	4
that of archimedes	4
can be done	4
would be to	4
followed by the	4
same ratio as	4
about the year	4
at least one	4
to set forth	4
used in the	4
with which it	4
is the mean	4
matter of fact	4
a plane as	4
and to show	4
of the tube	4
but we do	4
the four triangles	4
eudoxus of cnidus	4
the french mathematician	4
archimedes and his	4
prove that a	4
and it has	4
that they had	4
that the world	4
globe above us	4
add to the	4
much of this	4
to the world	4
the seven follies	4
claim that the	4
what has been	4
with the study	4
well to ask	4
diameter of a	4
the principles of	4
between two lines	4
same base and	4
well to call	4
corresponding proposition in	4
we have to	4
on the one	4
to elementary geometry	4
to inscribe a	4
the conclusion that	4
triangle that has	4
the shortest line	4
with which the	4
c without violating	4
proposition about the	4
to mean the	4
to be made	4
be applied to	4
of the value	4
may think of	4
seem to have	4
is to say	4
about one hundred	4
of the exercises	4
of two circles	4
philippus of mende	4
degree of accuracy	4
geometry was taught	4
that this is	4
the teacher and	4
geometry is studied	4
arc of a	4
is the limit	4
may be described	4
quadrant of the	4
is no longer	4
all right angles	4
limit of a	4
a sphere is	4
shown by the	4
per cent of	4
means of which	4
it to the	4
and in particular	4
in contact with	4
determine a straight	4
of the whole	4
that given by	4
in the past	4
boys and girls	4
or it may	4
there was a	4
the motion of	4
if p is	4
the best way	4
is claimed that	4
straightedge and compasses	4
cos o x	4
that is a	4
line and a	4
that the ancients	4
measure of the	4
number of simple	4
history of geometry	4
and the one	4
triangle are equal	4
area of an	4
the first part	4
the father of	4
lines are cut	4
we have no	4
points of the	4
in geometry that	4
or universal solvent	4
that are not	4
of geometry to	4
with most of	4
a straight angle	4
of the locus	4
have endeavored to	4
a book that	4
may be of	4
long before the	4
it is with	4
has already been	4
ginn and company	4
the thirteenth century	4
the best known	4
this is easily	4
the efforts of	4
should be a	4
that of euclid	4
while it is	4
academy of sciences	4
is said that	4
euclid does not	4
of the compasses	4
it in the	4
by those who	4
when we say	4
a point is	4
shown in fig	4
is open to	4
right angles are	4
of the four	4
and the elixir	4
of a similar	4
the second part	4
same is true	4
than the one	4
among the greeks	4
is the most	4
twice the volume	4
seen from the	4
on the line	4
those who are	4
bisects the circle	4
the most difficult	4
this is done	4
point of intersection	4
but the proof	4
that the line	4
the arrangement of	4
is merely a	4
the corners of	4
the exterior angles	4
an egg more	4
we may use	4
the cylinder is	4
boy or girl	4
this form of	4
the second and	4
by the name	4
circle and the	4
preliminary to the	4
of the discovery	4
they are so	4
a little of	4
interest in the	4
to lie on	4
from the arabic	4
and that of	4
of the incommensurable	4
circle is a	4
civics and health	4
be done by	4
distant from the	4
shall not be	4
in the curriculum	4
we can find	4
the invention of	4
proclus speaks of	4
the direction of	4
of the equation	4
this includes the	4
the postulate of	4
familiar with the	4
is not as	4
the reductio ad	4
any triangle the	4
of geometry and	4
determined by n	4
in the east	4
of double the	4
a pupil to	4
a definition of	4
been made in	4
and so for	4
is drawn through	4
band of sponge	4
of one hundred	4
be said that	4
the dihedral angle	4
of the proof	4
in addition to	4
true of the	4
a very early	4
such a simple	4
simple matter to	4
be noted that	4
be used in	4
the semicircle is	4
of the octahedron	4
they are more	4
the same idea	4
it is easier	4
about the same	4
used by the	4
brought into contact	4
he was a	4
was the first	4
diameter is given	4
of its sides	4
speak of a	4
of any other	4
equivalent to a	4
have been a	4
is the first	4
the ashes of	4
draw a line	4
not to be	4
but it should	4
might have been	4
line as the	4
suggested by the	4
teacher of geometry	4
the converse of	4
know that the	4
that has one	4
the polyhedral angle	4
to the figure	4
limit of the	4
the powers of	4
the same direction	4
out of all	4
as is the	4
which it was	4
which have the	4
states that the	4
basis for the	4
at least as	4
or even in	4
phase of the	4
one about the	4
to give the	4
the ahmes papyrus	4
the answer is	4
are not so	4
just as we	4
its angles a	4
be made by	4
in the mensuration	4
all such cases	4
triangle that which	4
of these impressions	4
line parallel to	4
a short time	4
of his work	4
the analogous proposition	4
may be taken	4
the accepted ratio	4
is customary to	4
in this form	4
quadrature of the	4
by which a	4
and a circle	4
plane parallel to	4
an error of	4
thousandth of an	4
it is made	4
to be understood	4
those on the	4
of book i	4
been known to	4
it is called	4
any other subject	4
cut the circle	4
on the teaching	4
the comprehension of	4
the whole is	4
to move in	4
in relation to	4
since it is	4
of straight line	4
is such that	4
and that for	4
to see how	4
found to be	4
give only the	4
an opportunity for	4
or any other	4
thing that is	4
and it may	4
could have been	4
fifth century b	4
find that the	4
equal in area	4
of book iv	4
the manner of	4
in the book	4
included side of	4
possible to find	4
commentary on euclid	4
of a parallelogram	4
the reasons for	4
which he could	4
us that he	4
having the same	4
so simple that	4
the fusion of	4
class to have	4
a few illustrations	4
the circumference to	4
the advantage of	4
of the screw	4
of three dimensions	4
a point equidistant	4
is impossible to	4
the reason is	4
to the point	4
those of the	4
equal circles equal	4
the making of	4
it is too	4
made to move	4
like the following	4
the cutting plane	4
in the high	4
to make an	4
to be considered	4
straight line drawn	4
the one about	4
illustrations of the	4
into three equal	4
if we say	4
has been studied	4
and half an	4
proved that the	4
of the mensuration	4
that they have	4
who have studied	4
two lines are	4
it is entirely	4
same time we	4
a certain point	4
of capillary attraction	4
the subject matter	4
of the water	4
to us from	4
the bible and	4
division of the	4
have to do	4
of n sides	4
in a triangle	4
or at least	4
as to form	4
in the textbook	4
it is so	4
shows that the	4
showing that the	4
the plumb line	4
the regular pentagon	4
a good deal	4
to two sides	4
of our modern	4
feet from the	4
into contact with	4
equal and parallel	4
circle of which	4
piece of work	4
opposite the equal	4
when we come	4
the state of	4
needless to say	4
to introduce this	4
this problem is	4
hole in the	4
made to coincide	4
a broken line	4
of these terms	4
as it was	4
of the tree	4
theory of proportion	4
that he should	4
the rule of	4
to the sun	4
under the microscope	4
it is difficult	4
for the case	4
a proof is	4
on a circle	4
a correspondent of	4
to the first	4
there is one	4
number of faces	4
base of the	4
to the mind	4
of a proposition	4
of the case	4
two points determine	4
have come down	4
b and d	4
of geometry by	4
as to its	4
is given in	4
nevertheless it is	4
of constructing a	4
at the most	4
are found in	4
inscribe a regular	4
to the needs	4
lines are parallel	4
does not seem	4
reduce the number	4
of a very	4
to the great	4
on the opposite	4
and any given	4
and others of	4
through the vertex	4
of a cone	4
a man who	4
so that in	4
is on the	4
of pure geometry	4
it is customary	4
four for the	4
at its extremity	4
for a class	4
how to attack	4
is also the	4
invented a perpetual	4
by the french	4
angles a right	4
which shall be	4
between axiom and	4
if they had	4
illustration for example	4
the same line	4
of that of	4
of these is	4
is a straight	4
to see that	4
by taking a	4
on this subject	4
about a point	4
than a right	4
so that their	4
well known to	4
the first three	4
the most notable	4
any one of	4
of the faces	4
and the three	4
of plane figures	4
and do not	4
should be so	4
applications of geometry	4
the bisector of	4
areas of the	4
of a segment	4
in the works	4
as the bounding	4
that the angle	4
it is that	4
are not in	4
perp to x	4
point equidistant from	4
the fixation of	4
henrici and treutlein	4
it is often	4
the diameter is	4
any part of	4
third century a	4
if the proposition	4
the triangle and	4
copy of the	4
which has been	4
few of the	4
become a millionaire	4
de morgan says	4
to use a	4
is sufficient to	4
the mind of	4
illustration in the	4
portion of the	4
a solution of	4
has always been	4
acute angle is	4
there is nothing	4
let us suppose	4
any kind of	4
a feature that	4
tangent to the	4
be inscribed in	4
to the front	4
a given straight	4
teachers of geometry	4
other as the	4
where there is	4
proofs of the	4
fall a perpendicular	4
be made of	4
hope that the	4
the strength of	4
angle on the	4
that it would	4
and his fulcrum	4
if there is	4
less than one	4
bisectors of the	4
of the crucible	4
is more than	4
in the illustration	4
how to find	4
from an external	4
are added to	4
the limits of	4
a line perpendicular	4
in the art	4
that the whole	4
comes to the	4
stated in the	4
the treatment of	4
be understood that	4
of the ordinary	4
taken from the	4
the distance from	4
at a very	4
any point on	4
the space inclosed	4
he says that	4
in the schools	4
time we must	4
which might be	4
definitions of geometry	4
the center is	4
as we know	4
may be obtained	4
and to the	4
been called the	4
two of the	4
the circumference and	4
the list of	4
line from a	4
respect to the	4
now in the	4
to improve the	4
these lines is	4
for a time	4
as an exercise	4
the quadrature of	4
to state the	4
to the study	4
it is desirable	4
is a simple	4
good deal of	4
the ability to	4
for the first	4
is required to	4
they can be	4
the other is	4
of a great	4
and for the	4
flourished about b	4
are subtracted from	4
the interest of	4
by no means	4
shown in this	4
in modern times	4
the incommensurable case	4
with one of	4
seen that the	4
the price of	4
definition is not	4
an arc of	4
prior to the	4
are not the	4
end in view	4
let it be	4
us suppose that	4
sense of touch	4
may be the	4
drawn from a	4
made up of	4
that is so	4
be used as	4
taught in the	4
the base is	4
two propositions are	4
to the eye	4
the development of	4
will now be	4
is found that	4
precisely the same	4
why do we	4
equals are added	4
thought that the	4
to know the	4
to form a	4
a space of	4
before his time	4
in any case	4
are needed in	4
be observed that	4
usually given in	4
or in equal	4
the plan of	4
a machine which	4
most of our	4
it is claimed	4
of the famous	4
geometry as a	4
referring to the	4
the vertices of	4
at this stage	4
be required to	4
the elements of	4
the solution is	4
for such a	4
think of the	4
as for the	4
the problem to	4
arrangement of the	4
at once to	4
the pleasure of	4
and such a	4
point the hour	4
compasses and straightedge	4
in most of	4
looked upon as	4
writings of the	4
with any given	4
by the class	4
the centers of	4
and to have	4
of a subject	4
injected into the	4
which they are	4
the subject and	4
was not a	4
there was no	4
as the extremity	4
to be an	4
description of the	4
if it should	4
who does not	4
added to the	4
pupils in the	4
to calculate the	4
greater than that	4
does not use	4
from the study	4
he would not	4
the perimeters of	4
was to be	4
of the pyramid	4
of the point	4
is easily shown	4
the capacity of	4
the term is	4
no one can	4
and the proposition	4
that should be	4
who did not	4
appear to be	4
in the british	4
show that the	4
of the beginner	4
in a country	4
was that of	4
knowledge of the	4
either directly or	4
axiom and postulate	4
of two sides	4
the introduction to	4
necessary for the	4
which has its	4
the proposition relating	4
the watch is	4
of the figures	4
a hole in	4
if they were	4
does not make	4
with which to	4
and to give	4
of the period	4
leonardo of pisa	4
in the hands	4
may be asked	4
a copy of	4
has long been	4
the two triangles	4
piece of land	4
centers of the	4
statement that a	4
in every case	4
to the circle	4
be seen from	4
a type of	4
in the fact	4
a to b	4
and circumscribed polygons	4
referred to in	4
of the sun	4
of simple exercises	4
at least to	4
they must have	4
of our present	4
as the following	4
with which we	4
as one of	4
the power of	4
to be that	4
motion of the	4
in the line	4
propositions of euclid	4
angles of all	4
to a good	4
a part of	4
not until the	4
a given cube	4
the mouse or	4
we shall have	4
on a square	4
to two angles	3
a period of	3
of any of	3
that comes from	3
the sources of	3
two intersecting lines	3
the problem becomes	3
drawing of a	3
we cut through	3
it is manifestly	3
to distinguish between	3
knows that the	3
of lincoln cathedral	3
and whose altitude	3
questions like the	3
is the square	3
a given distance	3
was not until	3
the parts of	3
is a special	3
it is used	3
as has been	3
the section is	3
all the straight	3
given point in	3
remains the same	3
subject in the	3
accepted by the	3
l a is	3
to the two	3
this axiom is	3
results which have	3
of the oblique	3
the tree is	3
needs of the	3
of euclid are	3
and the compasses	3
the values of	3
the other also	3
translation from the	3
in the subject	3
are not used	3
the geometric converse	3
with the modern	3
accuracy with which	3
the one in	3
and of course	3
make for better	3
a working model	3
is made by	3
of the senses	3
the case is	3
no apology for	3
the educational standpoint	3
which he claimed	3
this connection that	3
by a straight	3
length of a	3
or even of	3
view of these	3
in writing of	3
that a line	3
also true of	3
upon his tomb	3
a circle a	3
moving railway carriage	3
the work and	3
the rectangular parallelepiped	3
in which they	3
but let us	3
that dealing with	3
the angle of	3
of a tree	3
tangents to the	3
small budget of	3
history of greek	3
lead to the	3
amount of time	3
the essential features	3
conscious of it	3
parallel to ab	3
take away the	3
as the number	3
of a trapezoid	3
well as the	3
the lower end	3
have long been	3
they had discovered	3
principle of continuity	3
assigned to the	3
and natural philosophy	3
with a little	3
than in the	3
hung a foot	3
given for the	3
into elementary geometry	3
are sufficient to	3
a clock which	3
the intercepted arcs	3
a sort of	3
and the rest	3
that the ordinary	3
the angle between	3
to move the	3
parallel lines are	3
of many of	3
we take the	3
to mean a	3
point from which	3
at liberty to	3
a triangle whose	3
placed on a	3
we must not	3
one of our	3
is determined by	3
it is of	3
the british association	3
more value than	3
it with the	3
variations of this	3
of some of	3
a translation of	3
of these forms	3
space of three	3
a point p	3
have claimed that	3
search for the	3
seven wise men	3
the greater side	3
to assist in	3
angle by taking	3
exterior angles of	3
quadrant used for	3
says that the	3
in america to	3
some idea of	3
of the pythagorean	3
for a perpetual	3
the surface is	3
the first one	3
to be taught	3
the difference of	3
he had been	3
the alkahest or	3
the whole subject	3
of the intercepted	3
surface which is	3
his first proposition	3
the class in	3
use of these	3
idea of a	3
of the natural	3
worth the while	3
fascination of the	3
it on the	3
it would take	3
is described in	3
of the day	3
and just as	3
he is said	3
so it is	3
a considerable amount	3
of such work	3
to the radius	3
tangent to a	3
a regular hexagon	3
may lead to	3
of this axiom	3
a line of	3
pythagoras discovered the	3
equal straight lines	3
what is called	3
of the leading	3
lines without breadth	3
in the sense	3
only the ability	3
from two given	3
such a thing	3
here is a	3
the right hand	3
may be moved	3
by the italian	3
and that they	3
explain what is	3
be interested in	3
intercept equal arcs	3
double of the	3
equilateral triangle and	3
if two straight	3
of the picture	3
of a curve	3
propositions of solid	3
it for the	3
and it seems	3
be the case	3
so long as	3
geometry it is	3
the two sides	3
and bisected angles	3
ordinary paper protractor	3
of the side	3
tenths of an	3
the level of	3
the ancient geometry	3
the naked eye	3
on a straight	3
bear in mind	3
his perpetuum mobile	3
in motion by	3
to grasp the	3
any one to	3
recreations and problems	3
sacrificed an ox	3
some form of	3