Mathematics

'There is no certainty where one can neither apply any of the mathematical sciences nor any of those which are based upon the mathematical sciences!

As I have shown, here at the side [diagram], various ways of squaring the circles, that is by forming squares of a capacity equal to the capacity of the circle, and have given the rules for proceeding to infinity, I now begin the book called 'De Ludo Geometrico', and I give also the method of the process to infinity. c.a. 45 v. a

A body is something of which the boundaries form the surface.

The surface is not part of the body nor part of the air or water that surround it, but it is a common boundary .... in which the body ends in contact with the air, and the air in contact with the . . . c.a. 91 v. a

What is that thing which does not give itself, and which if it were to give itself would not exist?

It is the infinite, which if it could give itself would be bounded and finite, because that which can give itself has a boundary with the thing which surrounds it in its extremities, and that which cannot give itself is that which has no boundaries. c.a. 131 r. b

Surface is the touching-part [contingenzia] of the extremities of bodies, that is it is made by the extremities of the body of the air, together with the extremities of the bodies which are clothed by this air, and it is that which completes and forms with this air the boundary of the bodies surrounded by the air, and completes this air with the bodies clothed by it, and it does not participate either in the body which surrounds it or in that which is surrounded by it. It is rather the true common boundary of each of these, and it is that which divides the one body from the other, as one may say the air or the water from the body that is enclosed in these. c.a. 182 r. a

Arithmetic is a mental science and forms its calculations with true and perfect denomination; but it has not the power in its continuing quantities which irrational or surd roots [radici sorde] have, for these divide the quantities without numerical denominaton. c.a. 183 v. a

Surface is a flat figure which has length and breadth and is uniformly without depth. c.a. 246 v. b

A point is not a part of a line. Tr. 63 a

[ With drawing}
\ To ascertain the width of a river]

If you would ascertain the exact distance of the breadth of a river proceed as follows : — plant a staff upon the river bank at your side and let it project as far from the ground as your eye is from the ground; then withdraw yourself as far as the span of your arms and look at the other bank of the river, holding a thread from the top of the staff to your eye, or if you prefer it a rod, and observe where the line of sight to the opposite bank meets the staff. b 56 r.

[With drawing] [A level resting on a support from the base of which there is a cord to its ends]

This is the way that the level should be made : that is it is two braccia long, an inch thick, and square; and it should be of pine so that it may not twist, and have in the top of it a groove of the thickness of a finger and of the same depth. Then moisten the cord and fill the groove with water, and lower first the one end and then the other until the water stands level with the sides. Then proceed to wipe away with the finger the water that flows over the ends of the groove until these become dry, and fix two pieces of iron at m n, of the thickness of the cord, and see that one fastens the other and the thing seen. b 65 v.

A thing which moves acquires as much space as it loses. e 7 v. and 25 v.

OF MECHANICS

Mechanics is the paradise of the mathematical sciences because by means of it one comes to the fruits of mathematics. e 8 v.

OF THE SQUARING OF THE SURFACE OF A SPHERE WITH STRAIGHT MOVEMENT

The knowledge of the aliquot part gives knowledge of its whole; whence it follows that the squaring of the eighth part of the surface of a sphere gives knowledge of what is the square of the whole of this sphere; and let this be the knowledge of the eighth of the sphere: a b c.

Second figure. In the second figure c d e one divides the eighth part of the spherical surface in parallels of equal breadth and straightens the curve of the two sides c d and d e; this is done with movement upon a level place.

Third figure. In this third figure there is that which was promised in the second, and the straightened sides / g and g h are all the parallels of the second, which are enlarged and elongated by means of their movement, because there are the same number of parallels made upon the extended lines / g and g h (which are equal to each other) ; the whole being increased the parts also have increased.

Fourth figure. In the fourth figure one makes equal pyramidal divisions as shown in / g h.

Fifth figure. In the fifth figure the points of the pyramids are opened and enlarged, the same number of pyramids are reproduced, and the square n m o p is formed; but first by movement one straightens the line i /, and one has the fourth part of the spherical surface.

The junction of the curves c d e straightened at / g h forms a rectangle. e 24 r.

DEFINITION OF HELIX

A helix is a single curved line the curve of which is uniformly irregular and it goes revolving round a point at a distance uniformly irregular.

DEFINITION OF HEMISPHERE

A hemisphere is a body produced by a half sphere contained by the circle and the surface of the half sphere.

The movement of the hemisphere commenced by the circumference of its greatest circle ends in the middle of this hemisphere, after having described a spiral curve.

This is proved by the second concerning compound impetus which says: 'Of compound impetus one part will be as much slower than the other as it is shorter', and: 'That will be shorter which is farther distant from the direct line of the movement made by its mover'. Therefore the movement of the hemisphere being made up of the movement of many whole revolutions is of the same movement as a half revolution.

MENSURATION

When you wish to measure the breadth of a river withdraw from its bank to a somewhat greater distance than the width of the stream and observe some fixed mark on the opposite bank of the river. Let a b represent the width of the river, and a c the space to which you withdraw from the river, this being somewhat longer than the width of the river.

Next draw at the end of this distance a perpendicular line of whatever length you please, and let this be the line c d.

And from this [spot] d observe again the mark b, which you noted
on the opposite side of the river, and make a mark f 1 upon the [op-
posite] bank at the point which is in the same line d b. After having
done this bisect the perpendicular line c d at the point e and from this
point e make another perpendicular line at exact right angles, and
make a mark where it intersects the line d f 1 , and from this make the
third perpendicular line g f 2 . You will thus have formed the quad-
rilateral c f e g, of which you know that the side c f is equal to f b,
because as the point e is in the centre of the line c d so the point f is in
the centre of the other line c b\ then take a f (from the bank) from
f c, that is f h, and you have remaining h c, a distance equal to the
width of the said river. e 51 v.

All the pyramids made upon equal bases in parallel spaces are equal
to each other.

The greatest pyramid that can be drawn from a cube will be the
third of the whole cube. e 56 r.

The intercentric line is said to be that which starts from the centre
of the world and which rising therefrom in one continuous straight line

616 MATHEMATICS

passes through the centre of the heavy substance suspended in an in-
finite quantity of space. e 69 r.

OF THE FIVE REGULAR BODIES

Against some commentators who blame the ancient inventors from
whom proceed the grammars and the sciences and campaign against
the dead inventors, and why they have not discovered through idleness
how to become inventors themselves, and how with so many books they
set themselves continually to confute their masters by false arguments :

They say the earth is hexahedral, 1 that is to say cubical, that is to say
a body with six bases, and they prove this by saying that there is not
among regular bodies a body of less movement or more stable than the
cube. And they attribute to fire the tetrahedron, that is the pyramidal
body, this being more mobile according to these philosophers than the
earth; for this reason they attribute the pyramid to fire and the cube to
the earth.

Now if one had to consider the stability of the pyramidal body and
to compare it with that of the cube, this cube is without any compari-
son more capable of movement than the pyramid, and this is proved
as follows:

The cube has six sides, the regular pyramid four, and these are
placed here in the margin at a b\ a is the cube, b the pyramid. In order
to define this proof I will take a side of the cube and a side of the
pyramid which will be c d; I maintain that the cube c will be more
adapted to a movement of circumvolution than the pyramid d. And
let e f, below, represent the commencement of these movements. I say
that as a matter of fact if the base of the cube and the base of the pyra-
mid rest upon the same plane the pyramid will turn the third of its
bulk to fall upon its other side, and the cube will turn the fourth part
of its circuit to change the other side in order to make a base. From
these two demonstrations the conclusion follows that the cube will turn
completely with the change of its four sides upon the same plane, while
the triangle of pyramid will turn completely with three of its sides
upon the same plane. The pentagon turns all its five sides and so the
more sides there are the easier is the movement because the figure ap-

1 MS. has tetracedronica coe cubica — presumably a slip of the pen.

MATHEMATICS 617

proaches more nearly to a sphere. I wish it to be inferred therefore that
the triangle is of slower movement than the cube and that therefore
one should take the pyramid and not the cube for the earth.

f 27 v.

OF PROPORTION

If from two like wholes there be taken away like parts there is the
same proportion between part and part as there is between whole and
whole.

It follows that if of these two circles the one is double the other, the
quarter portion of the larger is double the quarter portion of the
smaller.

And there is the same proportion between one remainder and the
other as between one whole and the other.

And the same proportion between part and part as there is between
remainder and remainder.

When two circles touch the same square at four points one is double
the other.

And also when two squares touch the same circle at four points one
is double the other. g 17 r.

GEOMETRY

The circle that touches the three angles of an equilateral triangle is
triple the triangle that touches the three sides of the same triangle.

The diameter of the largest circle made in the triangle is equal to
two thirds the axis of the same triangle. g 17 v.

The proportion of circle to circle is as that of square to square made
by the multiplication of their diameter by itself. Now make two squares
in such proportion as pleases you, and then make two circles, of which
one has for its diameter the side of the greater square, and the other has
for its diameter the side of the lesser square.

Thus by the converse of the first proposition you will have two circles
which will bear the same proportion one to another as that of the two
squares. g 37 r.

618 MATHEMATICS

TO OBTAIN THE CUBE OF THE SPHERE

When you have squared the surface of the circle divide the square
into as many small squares as you please, provided that they are equal
one to another, and make each square the base of a pyramid, of which
the axis is the half diameter of the sphere of which you wish to obtain
the cube; and let them all be equal. g 39 v.

[Circles and squares]

Circles made upon the same centre will be double the one of the
other, if the square that is interposed between them is in contact with
each of them. And double the one of the other will be the squares
formed upon the same centre, when the circle that is set in between
them touches both the squares.

This is proved because of the eight triangles of which the larger
square is composed the lesser square contains four.

There is the same proportion between circle and circle that there is
between square and square, formed by the multiplication of their
diameters.

Of all the parts of circles which may be in contact inside a right
angle the greater is always the equal of all the less; and of all the par-
allels which receive these parts in themselves the greater always con-
tains and is the equivalent of all the small parallels formed in this right
angle a b c. g 40 r.

DEFINITION OF FOUR GROUPS OF PARALLELS

Parallel figures are of four kinds. The first is enclosed between two
straight and equidistant lines; the second is between two equidistant
lines of uniform curve; the third is between two equidistant lines of
varying curve, such as the parallel lines made around the centre of the
circle; the fourth is formed of a single line curved round a point at an
equal distance, that is the line of the circumference round the centre
of its circle.

And all these lines are of uniform nature since with movement the
straight line becomes curved and the curved line becomes straight, by

MATHEMATICS 619

means of the impressions of the straight planes upon the curved and of
the curved upon the straight.

By one of the 'Elements' [of Euclid].

All the rectilinear triangles made upon equal bases and between
parallel straight lines are equal to one another. c 59 r.

If from unequal things there be taken away equal parts the re-
mainders will be unequal; not in the former proportion but with a
greater excess of the greater quantity. G 69 v.

ARITHMETIC

Every odd number multiplied by an odd number remains odd.

Every odd number multiplied by an even number becomes even.

g 56 v.
[Of squaring the circle]

Animals that draw chariots afford us a very simple demonstration
of the squaring of a circle, which is made by the wheels of these
chariots by means of the track of the circumference, which forms a
straight line. g 58 r.

OF SQUARING THE CIRCLE AND WHO IT WAS WHO FIRST
HAPPENED TO DISCOVER IT

Vitruvius while measuring the mile by means of many complete
revolutions of the wheels that move chariots, extended in his stadia
many of the lines of the circumference of these wheels. He learnt these
from the animals that are movers of these chariots, but he did not
recognise that that was the means of finding the square equal to a
circle. This was first discovered by Archimedes the Syracusan who
found that the multiplication of half the diameter of a circle by half
of its circumference made a rectilinear quadrilateral equal to the circle.

g 96 r.

There is no certainty where one can neither apply any of the mathe-
matical sciences nor any of those which are based upon the mathemati-
cal sciences. g 96 v.

620 MATHEMATICS

That force will be more feeble which is more distant from its source.

H 7 I [23J V.

Every continuous and united weight which thrusts transversely rests
upon a perpendicular support.

If the weight is discontinuous and limited as when it is liquid or
granulated, it will make its thrust upon all sides, and making it thus
the pressure that is exerted upon the sides serves to lighten that upon
the foundations. h 74  r.

Should the contact which the thing united makes with the earth on
which it is supported be not in the line of its motive power, it will
prove heavier in proportion as it is farther distant from the line of its
motive power. h 113 [30 r.] v.

The heaviest part of every body that is moved will be the guide of
its movement. h 115 [28 r.] v.

Similarity does not imply equality. 1 16 r.

The fact that a thing may be either raised or pulled causes great
difference of difficulty to its mover; for if it is a thousand pounds and
one moves it by simply lifting it it shows itself as a thousand pounds,
whereas if it is pulled it becomes less by a third; and if it is pulled with
wheels it is diminished by as many degrees in proportion to the size of
the wheel, and also according to the number of the various wheels.
And with the same time and power it can make the same journey,
with different degrees of time and power also in the same time and
movement; and it does this merely by increasing the number of the
wheels, on which rest the axles which would also be increased.

1 17 r.

By the ninth of the second of the Elements, which says that the
centre of every suspended gravity stops below the centre of its support,
therefore : —

The central line is the name given to what one imagines to be the
straight line from the thing to the centre of the world.

The centre of all suspended gravity desires to unite with the central
line of its support.

MATHEMATICS 621

And thai suspended gravity which happens to be farther removed
from the central line of its support will acquire more force in excess of
that of its natural weight.

Now in conclusion I affirm that the water of the spiral eddy 1 gives
the centre of its gravity to the central line of its pole, and every small
weight that is added on one of its sides is the cause of its movement.

1 22 v.

\T/ie Wonders of Mechanics]
Rule \ Diagram]

Pivots of the greatest force serve for the movements that go and
return such as those of bells, saws and things of the same nature.

A pound of force at b has for result at m ten thousand thousands of
millions of pounds, and the figure opposite does the same, being of the
same nature and only differing in that the wheels are whole as they
have to turn always in a single direction. And know that when the
first above gives a hundred thousand thousands of millions of turns,
that below only gives one complete turn.

These are the wonders of the science of mechanics.

In this manner one may make a bell to swing on a pivot so that it
will be sounded by a slight wind, the bell having its opposite weights
equal and equidistant from its centre. 1 57  v.

[Diagram]

This arrangement will produce a revolving movement of such dura-
tion that it will appear incredible and contrary to nature, because it
will make much movement after that of its mover. And it causes the
weight m to fall from such a height that the wheel gives thirty revo-
lutions and more, and then remains free after the manner of a spinning
top; and in order to avoid noise this stone ought to fall upon straw.

And to make one wheel greater than another down in succession
the one below the other, is only necessary in order that the rim of the
wheel below may not stop and impede the pivot of the other.

1 58  r.

1 MS. dele uiti.

622 MATHEMATICS

PROPORTION IN ALL THINGS

Proportion is not only found in numbers and measurements but also
in sounds, weights, times, positions, and in whatsoever power there
may be. k 49 [48 and 15] r.

How one of Xenophon's propositions is incorrect:

If unequal things are taken away from unequal things and these are
in the same proportion as the first inequality, the remainders will have
the same proportion in their inequality. But if from unequal things
equal things are taken away the remainders will still be unequal, but
not in the same proportion as before.

Consider these examples: in the first place let the parts taken away
be in the same proportion as the wholes, that is let 2 and 4 stand for
the two wholes so that the one is double the other. Then take 1 away
from 2, there remains 1; take 2 away from 4, there remains 2; and
these remainders have the same proportion as the wholes and as the
parts taken away. Therefore if 1 be taken from 2 and 2 from 4 there
remains the same proportion as at first, that is 1 and 2 which is double
as I said before : it would follow that whoever should take away equal
things would change the former proportion; that is to say that if from
two numbers one of which is double the other such as 2 and 4 you
were to take away an equal thing, that is you took 1 from 2 and 1
from 4, there would be left 1 and 3, that is numbers of which one
would be three times the other and therefore more than double in
difference.

You therefore, Xenophon, who wished to take away equal parts
from unequal wholes, believing that although the remainders were
unequal they were still in the same proportion as at first, you were
deceiving yourself! k 61  r. and v.

DEFINITIONS OF A STRAIGHT LINE

First. A straight line is that of which each part finds itself of equal
height.

Second. A curved line is that which has a uniformly varying height
towards its extremities which are of equal height.

MATHEMATICS 623

The first definition and the second are incorrect because a thing of
equal height must have every part of its bulk equally distant from the
centre of the world. So the curve / b o would be straight because it is
at a uniform distance from this centre, and the straight line a b c would
be curved, because every part of its length varies uniformly according
to the distance of the parts enclosed within extremities that are at equal
distance from the centre of the world.

And if you say that the straight line is that which receives three
points of equal height in its extent you still say wrong.

But if you say that a straight line is the shortest between two given
points you will give its true definition. k 78  v. and 79  r.

[With drawings]

The circle is the equal of a rectangular parallelogram made of the
fourth part of its diameter and the whole of its circumference, or you
may say of the half of its diameter and of its periphery (circumfer-
ence).

As though one were to suppose the circle e f to be resolved into an
almost infinite number of pyramids, and these being then extended
upon the straight line which touches their bases at b d and the half of
the height being thus taken away, so making the parallel abed, this
being precisely equal to the given circle e f.

With regard to the circumference of the circle it is desirable to
measure the quarter with a piece of bark of cane, in its spiral curve
and stretching it out, and to make a rule as to where is the centre of
the circle from which the movement of the extremity of the measure-
ment is directed, and similarly the centre of the movement of many of
its parts, and to make the general rule.

The circle is a parallel figure, because all the straight lines produced
from the centre to the circumference are equal and fall upon the line
of the circumference between equal angles and spherical lines. And the
same thing happens with the transversal lines of the parallelogram,
namely that they fall upon their sides between right angles.

All rectilinear pyramids, and those of curved lines formed upon the
same bases and varying uniformly as to the breadth of their length
between parallel lines of circumference, are equal.

K 79 [3 1 ] v - an d 80  r.

624 MATHEMATICS

Of pyramids of equal bases there will be found the same proportion
in the slopes of their sides as that of their heights. l 41 r.

Vitruvius says that small models are not confirmed in any operation
by the effect of large ones. As to this, I propose to show here below that
his conclusion is false, and especially by deducing the self-same argu-
ments from which he formed his opinion, that is by the example of
the auger, as to which he shows that when the power of a man has
made a hole of a certain diameter a hole of double the diameter
cannot then be made by double the power of the said man but by
much greater power. As to this one may very well reply by pointing out
that the auger of double the size cannot be moved by double the
power, inasmuch as the surface of every body similar in shape and of
double the bulk is quadruple in quantity the one of the other, as is
shown by the two figures a and n.
[Drawing] a n.

Here one removes by each of these two augers a similar thickness of
wood from each of the holes that they make; but in order that the holes
or augers may be of double quantity the one of the other they must be
fourfold in extent of surface and in power. l 53 r. and 53 v.

The right angle is said to be the first perfect among the other angles,
because it finds itself at the middle of the extremities of an infinite
number of other kinds of angles which differ from it, that is of an
infinite number of obtuse angles and an infinite number of acute angles,
and all these infinite angles being equal between themselves it finds
itself equidistant to each of them, being in the middle. m cover v.

THE THIRD LESSON OF THE FIRST

Triangles are of three kinds, of which the first has three acute
angles, the second a right angle and two acute angles, and the third
an obtuse angle and two acute angles.

The triangle with three acute angles may be of three different shapes
of which the first has three equal sides, the second two equal sides and
the third three unequal sides.

And the right-angled triangle may be of two kinds, i.e. with two
equal sides and with three unequal sides. m i r.

MATHEMATICS 625

The right-angled triangle with two equal sides is derived from the
half of the square. And the right-angled triangle with three unequal
sides is formed by the half of the long tetragon [rectangle |, and the
obtuse-angled triangle with two equal sides is formed by the half of
the rhombus cut in its greatest length.

The square is the name applied to a figure of four equal sides which
form within them four right angles, that is to say that the lines that
compose the angles are equal to each other. m 1 v.

LONG TETRAGON

The long tetragon [rectangle] is a surface figure contained by four
sides and four right angles; and although its opposite sides are equal
it does not follow from this that the sides which contain the right
angle may not be unequal between themselves.

The rhombus is of two kinds: the first is formed by the square and
the second by the parallelogram; the first has its opposite angles equal
and likewise all its sides equal; its only variation consists in that no
side ends in equal angles but with an acute angle and an obtuse angle.

m 2 r.

RHOMBOID

The rhomboid is the figure that is formed from the rhombus, but
whereas the rhombus is formed from the square the rhomboid is
formed from the rectangle. It has the opposite sides and angles equal
to each other but none of its angles is contained by equal sides.

Parallel or equidistant lines are those which when extended con-
tinuously in a straight line will never meet together in any part.

m 2 v.

Every whole is greater than its part.

If [a thing] is neither larger nor smaller it is equal. m 3 r.

OF FIVE POSTULATES

That a straight line may be drawn from one point to another.
That with a centre it is possible to make a circle of any size.
That all right angles are equal to each other.

626 MATHEMATICS

When a straight line intersects two straight lines and the two angles
on one side taken together are less than two right angles these two
lines extended on this side will undoubtedly meet.

Two straight lines do not enclose a surface. m 6 r.

THE THIRD LESSON OF THE TENTH

Of the comparison made between the continuous and the definite
quantity, and how the continuous may have its parts communicating,
that is to say measured by a common measure as would be a measure
of one braccio, a measure that goes four times in a line of four braccia,
and then three in a length of three braccia; and so forms a unity which
enters four times in four numbers and also enters three times in three
numbers; and there is the same proportion between four braccia and
four numbers as there is between one number and one braccio.

m 6 v.

OF FIVE [SIX?] POSTULATES

The boundaries of the line are points, the boundaries of the surface
are lines and the boundaries of the body are surfaces.

That a straight line may be drawn from one "point to another.

And this line may also be extended as much as one pleases beyond
these points but the boundaries of this line will always be two points.

That upon the same point one may make many circles.

All right angles are equal to each other.

Parallel lines are those upon which if a transversal line be drawn
four angles are formed, which when taken within [on one side?]
equal two right angles. m 7 r.

If two squared surfaces have the same proportion to each other as
their squares, their sides will be corresponding, that is commensurable
in length.

And if there are two squared surfaces of which the sides are com-
mensurable in length it will follow that the proportion between them
will be as that of their squares.

And if the squared surfaces are not in the same proportion one to
another as are their squares, their sides will be incommensurable in
length. m 9 r.

\V1

MATHEMATICS 627

If two things are equal to a third they will be equal to one another.

m 13 r.

If from equal things one takes equal things away the remainders
ill be equal. M *5 r -

A straight line is that in which if one takes a point in any position
outside it, at such a distance that its length may share precisely such a
given line, and any straight line be drawn from the said point to each
of the said partitions, this line can be divided precisely in the same way
by each of these partitions.

Let us say that the line of which the proof has to be made is b /, that
the given point is a, that the space from the point to the extremity of
the line is a b: and that the lengths (partitions b, c, d, e, f each of itself
is equal to a b: I affirm that the line a c is double the space a b, and the
line a d is triple, a e quadruple and a f quintuple, m 13 v. and 14 r.

NINE PROPOSITIONS

The things which are equal to the same thing are also equal to each
other. And if to equal things one adds equal things the wholes will
still be equal.

And if from equal things one takes away equal things the remainders
will still be equal. And if from unequal things one takes away equal
things the remainders will be unequal. And if two things are equal to
another thing they will be equal to each other. And if there are two
things which are each the half of the same thing each will be equal to
the other. And if one thing is placed over another and touches it so
that neither is exceeded by the other these things will be equal to each
other. And every whole is greater than its part. m 16 r.

Geometry is infinite because every continuous quantity is divisible to
infinity in one direction or the other. But the discontinuous quantity
commences in unity and increases to infinity, and as it has been said
the continuous quantity increases to infinity and decreases to infinity.
And if you allow yourself to say that you give me a line of twenty
braccia I will tell you how to make one of twenty-one. m 18 r.

All the angles made round a point are together equal to four right
angles. m 31 v.

628 MATHEMATICS

[A man's leap]

If a man in taking a leap upon a firm spot leaps three braccia and
recoils as he takes his spring a third of a braccio, what would he lack
of his first leap?

And in like manner if it was increased by one third of a braccio how
much would he have added to his leap? m 55 r.

[Pyramids]

Multiply by itself the root of the number of this pyramid that you
wish and detach it towards some angle.

If I wanted the fourth part of the height of the base of this pyramid
which corresponds to the fourth part of the length of the pyramid I
should say : — four times four are sixteen, and so the piece removed will
be one sixteenth of the whole pyramid. And if you take away a part
such as the half of the base which corresponds to the half of the length
of this pyramid, you will say a half of a half is a quarter; therefore the
part taken away will be a quarter of the whole pyramid, and if you
multiply the three quarters of the base by the three quarters of the
length of the pyramid this will make nine sixteenths of the whole
pyramid.

This curved pyramid will find its end by finishing its circles. But if
such a pyramid were to go thousands of miles to unite itself you would
not be able to complete these circles; employ therefore the scale given.

m 86 v.
[Curved lines and pyramids]

If you cut above a section equidistant to the circle below making
use of its centre r, and if you cut below a section equidistant to the
circle above making use of its centre a, I wish to know if these two
lines be drawn with the same curve at what distance they will join,
or if they do not join where they will make their first approach and
how distant they will ever be and how near to each other.

I wish as regards two given lines which are curved to find whether
they are parallel or no, and if they are not parallel whether they are so
arranged as to form a pyramid or no, and if they ought to form a
pyramid at what fixed distance from the base their curved sides ought
to join.

And for this you will act as follows: detach a part between the base

MATHEMATICS 629

and the side and let this part be as great of such part of the base as of
the side, and the portion may be taken either from the part above or
the part below; and if you take it off the part above make it so that the
section may be equidistant to the circle below using its centre r, and if
you remove this portion from below make the section equidistant to
the line above using its centre a, and if you take away this portion
below make the section equidistant to the line above using its centre
0, and so continuing as the straight pyramid. m 87 r.

f Geo m etrical paradox ]

If the angle is the contact of two lines, as the lines are terminated in
a point an infinite number of lines may commence at this point, and
conversely an infinite number of lines may end together at this point:
consequently the point may be common to the beginning and the end
of innumerable lines.

And here it seems a strange matter that the triangle is terminated in
a point with the angle opposite to the base, and from the extremities of
the base one may divide the triangle into an infinite number of parts;
and it seems here that as the point is the common end of all the said
divisions the point as well as the triangle is divisible to infinity.

M 87 V.

The lines which form the circular parallels cannot be of the same
curve because as they complete their circles they will have their contact
or intersection in two places.

As regards the curve lines which have to make up the curved
parallels, it is necessary that the part and the whole of the one and the
part and the whole of the other should be together each of itself equi-
distant to a single centre. m 89 r.

SPHERICAL ANGLES EQUAL TO RIGHT ANGLES

Every four angles made within the circle of all the space of the circle
are equal to four right angles, whether the lines be curved and straight
or all straight or all curved.

Every quantity of lines that intersect at the same point will form as
many angles round this point as there are lines that proceed from it,
and these angles joined together will be equal to four right angles.

m 89 v.

630 MATHEMATICS

[With diagram]

Pelacani x says that the longer arm of this balance will fall more
rapidly than the shorter arm because in its descent it describes its quar-
ter circle more directly than the shorter arm does; and that since the
natural tendency of weights is to fall by a perpendicular line the more
the circle bends the slower will the movement become.

The diagram m n controverts this argument in that the descent of
the weights does not proceed by circles, and yet the weight of m the
longer arm falls.

When anything is farther away from its base it is less supported by
it; being less supported it keeps more of its liberty, and since a free
weight always descends, the extremity of the rod of the balance which
is most distant from the point of support, since it is heavy, will descend
of itself more rapidly than any other part. ms. 2038 Bib. Nat. 2 v.

[Levers]

In proportion as the extremity of the upper part of the balance ap-
proaches more nearly to the perpendicular line than the extremity of
the lower part, so much longer and heavier will the lower arm be than
the upper arm if the beam be of uniform thickness.

ms. 2038 Bib. Nat. 3 r.
[Suspended bodies]

The suspended body which is of smooth roundness will fall in the
line of its centre and will stop under the centre of the cord by which it
is suspended.

The centre of the weight of any suspended body will stop in a per-
pendicular line beneath the centre of its support.

ms. 2038 Bib. Nat. 3 v.

Gravity, force and material movement together with percussion are
four accidental powers in which all the visible works of mortals have
their being and their death.

Gravity is a certain accidental power which is created by movement
and infused into one element which is either drawn or pushed by an-
other, and this gravity possesses life in proportion as this element
strives to return to its former state. b.m. 37 v.

1 According to M. Ravaisson-Mollien the reference is to Biagio Pelacani of Parma
(born 1416), whom Tiraboschi calls filosofo e matematico insigne.

MATHEMATICS 631

The redness or yolk of the egg remains in the centre of the albumen
without sinking on either side, and it is either lighter or heavier or the
sanu weight as this albumen. If it is lighter it ought to rise above all
the albumen and remain in contact with the shell of the egg; and if it
is heavier it ought to sink down; and if it is of the same weight it
ought to be capable of remaining at one of the ends just as well as in
the centre or below it. b.m. 94 v.

The thing moved will never be swifter than its mover, b.m. 121 v.

The boundary of one thing is the beginning of another.
The boundaries of two bodies joined together are interchangeably
the surface the one of the other, as water with air. b.m. 132 r.

OF THE ELEMENTS

The bodies of the elements are united and in them there is neither
gravity nor lightness. Gravity and lightness are produced in the mix-
ture of the elements.

A point is that which has no centre.

A line is a length (extension) produced by the movement of a point,
and its extremities are points.

A surface is an extension made by the transversal movement of a
line, and its extremities are lines.

A body is a quantity formed by the lateral movement of a surface,
and its boundaries are surfaces.

A point is that which has no centre, and from this it follows that it
has neither breadth, length nor depth.

A point is that which has no centre, and therefore it is indivisible
from any angle and nothing is less than it is.

A line is a length made by the movement of a point, wherefore it
has neither breadth nor depth.

A body is a length and it has breadth with depth formed by the
lateral movement of its surface. b.m. 160 r.

[Definitions]

A point has no part; a line is the transit of a point; points are the
boundaries of a line.

632 MATHEMATICS

An instant has no time. Time is made by the movement of the
instant, and instants are the boundaries of time.

An angle is the contact of two lines which do not proceed in the
same direction.

A surface is the movement of a line, and lines are the boundaries of
a surface.

A surface has no body; the boundaries of bodies are surfaces.

b.m. 176 r.

A pyramidal body is that of which all the lines that proceed from
the angles of its base meet in a point.

And a body such as this may be clothed with an infinite number of
angles and sides.

A wedge-shaped body is one in which the lines that issue forth from
the angles of the base do not meet in one single point but in the two
points which end the line; and this ought not to exceed or fall short.

b.m. 176 v.

An instant has no time, for time is formed by the movement of the
instant and instants are the boundaries of time.

A point has no part.

A line is the transit of a point.

A line is made by the movement of a point.

Points are the boundaries of a line.

An angle is the contact of the extremities of two lines.

A surface is formed by the movement of a line moved sideways to
the line of its direction. b.m. 190 v.

[Propositions]

Every body is surrounded by an extreme surface.

Every surface is full of infinite points.

Every point makes a ray.

The ray is made up of infinite separating lines.

In each point of the length of any line whatever, there intersect lines
proceeding from the points of the surfaces of the bodies and [these]
form pyramids.

Each line occupies the whole of the point from which it starts.

At the extremity of each pyramid there intersect lines proceeding

MATHEMATICS 633

from the whole and from the parts of the bodies, so that from this
extremity one may see the whole and the parts.

The air that is between bodies is full of the intersections formed by
the radiating images of these bodies.

The images of the figures and colours of each body are transferred
from the one to the other by a pyramid.

Each body fills the surrounding air by means of these rays of its
infinite images.

The image of each point is in the whole and in the part of the line
caused by this point.

Each point of the one object is by analogy capable of being the
whole base of the other.

Each body becomes the base of innumerable and infinite pyramids.

That pyramid which is produced within more equal angles, will give
a truer image of the body from whence it is produced.

One and the same base serves as the cause of innumerable and
infinite pyramids turned in various directions and of various degrees
of length.

The point of each pyramid has in itself the whole image of its base.

The centre line of the pyramid is full of the infinite points of other
pyramids.

One pyramid passes through the other without confusion.

The quality of the base is in every part of the length of the pyramid.

That point of the pyramid which includes within itself all those that
start upon the same angles, will be less indicative of the body from
whence it proceeds than any other that is shut up within it.

The pyramid with the most slender point will reveal less the true
form and quality of the body from whence it starts.

That pyramid will be most slender which has the angles of its base
most unlike the one to the other.

That pyramid which is shortest will transform in greatest variety
the similar and equal parts of its base.

Upon the same quality of angles will start pyramids of infinite
varieties of length.

The pyramid of thickest point, more than any other will dye the
spot on which it strikes with the colour of the body from which it is
derived. b.m. 232 r.

634 MATHEMATICS

OF THE NATURE OF GRAVITY

Gravity is a fortuitous quality which accrues to bodies when they
are removed from their natural place.

OF THE NATURE OF LEVITY

Levity is allied with gravity as unequal weights are joined in the
scales, or light liquids are placed beneath liquids or solids which are
heavier than they . . . b.m. 264 r.

Take from one of five regular bodies a like body and so that what
is left may be of the same shape.

I wish to take a given pentagon from another pentagon and so that
the remainder may stay in the form of a pentagon, and they may be
bodies and not surfaces.

Reduce the given pentagon into its cube, and proceed thus with
the greater pentagon from which you have to extract the lesser; then
by the past rules take the lesser cube from the greater cube, and then
remake the pentagon from the remainder of this greater cube, which
by the aforesaid rules has remained cubed.

That which is here said of the cube is understood of all bodies which
touch the sphere with their angles, for what is made in the sphere may
be made in the cube. Forster 1 5 r.

All bodies have three dimensions, that is breadth, thickness and
length.

The changes and manipulations of bodies are six, namely shortening
and lengthening, becoming thicker and thinner, being enlarged and
compressed.

The surface has breadth and length and is uniformly devoid of
thickness.

The board is a flat body and has breadth, length and uniform
thickness.

Therefore when the board is of uniform thickness and its surface of
uniform quality we may use the table in all its manipulations and
divisions in the same manner and with the same rules as we use the
above mentioned surfaces. Forster 1 12 v.

MATHEMATICS 635

[Diagram \

The regular bodies are five and the number of those participating
between regular and irregular is infinite: seeing that each angle when
cut uncovers the base of a pyramid with as many sides as were the
sides of this pyramid, and there remain as many bodily angles as there
are sides.

These angles may be bisected anew and so you may proceed an
infinite number of times because a continuing quantity may be in-
finitely divided.

And the irregular bodies are also infinite through the same rule
aforesaid. Forster 1 15 r.

I will reduce to the form of a cube every rectangular body of equi-
distant sides. —

And first there will be a cylinder.

To get the square of a rectangular board that is longer than it is
wide according to a given breadth: ask yourself by how much its size
varies.

This may be done by the fifth of this, that is that I shall make of the
width or length of this board the cylinder of length equal to the said
width or length of the board, and then . . . Forster 1 31 r.

Geometry extends to the transmutations of metallic bodies, which
are of substance adapted to expansion and contraction according to the
necessities of their observers.

All the diminutions of cylinders higher than the cube keep the name
of cylinder. All the diminutions of the cylinder that are lower than the
cube are named boards.

The cube, a body of six equal sides contained by twelve equal lines
and eight angles of three rectangular sides and twenty-four right
angles; which body among us is called a die.

When you wish to treat of pyramids together as regards their in-
crease or diminution, and you treat of cyclinders, cubes or boards which
should be of the same height and breadth as these pyramids, then the
third of these bodies will remain in the said pyramid; and this you will
put concisely. Forster 1 40 v.

6^6 MATHEMATICS

METHODS OF MEASURING A HEIGHT

Let c f be the tower you wish to measure; go as far away from it as
you think desirable and take the range of it, as is shown in c b a, which
may be the length of an arm and half as high, and work it so that the
tower occupies the space b a; then turn the line b a along the level of
the ground, and it occupies as great a space of ground as it occupied
in height, and in the space of ground which it has occupied you will
find the true altitude of the tower. Forster i 48 v.

[Diagrams]

If a line falls perpendicularly upon another line it ends between two
right angles.

If a straight line falls upon another straight line and passes to the
intersection this intersection will stand in the middle of four right
angles.

If the two straight lines which intersect together between four right
angles shall have their four extremities equidistant to this intersection,
it is necessary that these ends be also equidistant from one another.

Forster 11 3 v.

If two circles intersect in such a way that the line of the circumfer-
ence of the one is drawn over the centre of the other as the other is of
it, these circles are equal, and the straight lines which pass from the
two points of intersection and from the centre to the other intersect
together within four right angles, and the circle made upon the two
centres will remain divided in four equal parts by such said intersec-
tion, and there will be made a perfect square. Forster 11 4 r.

If two three or four equal things are placed upon a thing which is
equal to them all, the part of the greater which protrudes will be equal
to the sum of the protruding parts of all the lesser ones; and the ex-
ample is the figure below. Forster 11 4 v.

ACTUAL PROOF OF A SQUARE

If four circles be so placed as to have their centres situated upon the
line of a single circle, in such a way that the line of the circumference

MATHEMATICS 637

of each is made over the centres of each, undoubtedly these will be
equal, and the circle where such intersection is made remains divided
in four equal parts, and it is in the proportion of a half to each of the
four circles, and within this circle will be formed the square with
equal angles and sides. Forster 11 5 v.

Every continuous quantity is divisible to infinity. Forster 11 53 v.

Gravity, force and accidental movement together with percussion,
are the four accidental powers with which all the visible works of
mortals have their existence and their end.

GRAVITY

Gravity is accidental power, which is created by movement and in-
fused in bodies standing out of their natural position.

HEAVY AND LIGHT

Gravity and lightness are equal powers created by the one element
transferred into the other; in every function they are so alike that for a
single power which may be named they have merely variation in the
bodies in which they are infused, and in the movement of their creation
and deprivation.

That body is said to be heavy which being free directs its movement
to the centre of the world by the shortest way.

That body is said to be light which being free flees from this centre
of the world; and each is of equal power. Forster 11 116 v.

Gravity, force, together with percussion, are not only interchangeably
to be called mother and children the one of the other and all sisters to-
gether, because they may be produced by movement, but also as pro-
ducers and children of this movement; because without these within us
movement cannot create, nor can such powers be revealed without
movement. Forster 11 117 r.

The accidental centre of the gravity that descends freely will always
be concentric with the central line of its movement, even though this
gravity should revolve in its descent. Forster 11 125 v.

638 MATHEMATICS

[Sketch]

a n forms the groove in the bank a quarter of a braccio on the inside,
by means of the grooves or teeth of iron, and these teeth rub against the
bases of the bank, and afterwards one seizes the handles of the rake,
and the soil that has collected upon it is placed in the box.

Forster in 18 r.
[Diagram]

That which is called centre is an indivisible part, and may more
readily be considered as round than of any other shape; therefore the
first part that surrounds it round is divisible whatever it may be; if it
be in the square beaten into a circle it enlarges. Forster in 26 v.

[Sketches]

The angle is terminated in the point; in the point intersect the images
of bodies. Forster in 29 v.

[Sketch]

WORM OF SCREW

The line b d ought to show how much this turns and similarly how
much the circle of the line a o turns, and take the number that is found
between the one number and the other; and upon this make your cal-
culation as is shown here below.
[S\etch\

m n is the line that finds itself between b d and a o, which you will
cause to take the direction as shown here below.
\ Sketch]

c r is the extent to which this line is slanting, that is the extent to
which the worm of the screw above turns over and drops.

Forster in 81 v.
[Sketch]

Multiply the line a o by the line o /?, and that which results multiply
with it that number of the parts of the half-diameter of the screw
which finds itself upon the length of the lever; and that which results
apportion it. Forster in 82 r.

And if you should only know the weight of the thing that you
wish to raise with the tackle and did not know how great weight or

MATHEMATICS 639

force was necessary in order to raise this weight, divide the number of
the pounds of your weight by the number of the wheels that there are
in the tackle, and that which comes out will be the uncertain weight
which will resist the certain with equal forces. Forster m 82 v.

If you wish with certainty to understand well the function and the
force of the tackle, it is necessary for you to know the weight of the
thing that moves or the weight of the thing moved; and if you would
know that of the thing that moves multiply it by the number of the
wheels of the tackle, and the total that results will be the complete
weight which will be able to be moved by the moving thing.

Forster in 83 r.

Such proportion will the weight have which is suspended by means
of the lever through the cord of the windlass to the force that the
mover exerts for its suspension, as has the half of the diameter of the
windlass to the space that is found upon the lever, between the hand of
its mover and the centre of the thickness of the said windlass.

Forster in 83 v.
[Sketch]

If you multiply the number of the pounds that your body weighs by
the number of the wheels that are situated in the tackle you will find
that the number of the total that results will be the complete quantity
of pounds that it is possible to raise with your weight.

Forster in 84 r.

That body of which the parts that are enclosed between the surface
and the centre are equal in substance, weight and size, if it be sus-
pended transversely by its opposite extremities will give an equal part
of its weight to its supports.

That wheel of which the centre of the axis is the centre of its circle,
will in all circumstances perform its functions in perfect balance; and
equal bodies suspended from the opposite extremities of its circle will
stand in equal counterpoise the one to the other. Forster in 84 v.

[Sketch — tackle]

It can be so made that although the counterpoises are different in
weight the one to the other in equal arms of balances, they stand at
equal resistance the one to the other : see in the instrument represented

640 MATHEMATICS

in the equal arms of the upper balances, sixteen [pounds] weight be-
low stands in resistance to eight. Forster in 85 r.

In proportion as the number of the wheels is greater so will the fall
of the counterpoise be greater than the rise of the greater weight.

In proportion as the number of the wheels is greater so will the num-
ber of the arms of the cord collected by the windlass be greater than
that of the weight that is raised. Forster in 85 v.

The pulling of the tackle requires force, weight, time and movement.

OF THE MOVEMENT OF THE CORDS

As many as may be the number of the wheels of the tackle so much
will the cord be swifter in its first movement than in its last.

OF THE WEIGHT

In proportion to the number of the wheels so much will the weight
sustained be greater than that which supports it. Forster in 86 r.

[Sketch. 'Cord of the windlass.' 'Multiply that weight by the number

of the wheels.']

If you wish to ascertain how much cord a windlass will collect after
it has passed through the whole or as few as two [turns] of a tackle of
four wheels, know that for every braccio that the weight is raised, the
windlass will collect four [braccia] by the four wheels of the tackle;
and if the wheels were twenty, for every braccio that the weight was
raised the windlass would need two braccia of cord.

In the raising of the weight the windlass would need as many times
more braccia of cord than the weight would raise, according to the
number of the wheels which are collected in the tackle.

Forster in 86 v.

If the wheels are two and you wish to raise the weight one braccio
the windlass collects two braccia; the proof is this: let us say n m is
one braccio, and so n f may be another; let us say that I wish to raise
the weight m one braccio: it is evident that the cord n m f which is

MATHEMATICS 641

two braccia will be no more in its position and the windlass will gather
up as much again.

In proportion to the number of the wheels that move in the tackles
by so much will the cord of the first movement be swifter than that of
the last. Forster in 87 r.

DEFINITION OF THE NATURE OF THE LINE

The line has not in itself any matter or substance but may more
readily be called an incorporeal thing than a substance, and being of
such condition it does not occupy space. Therefore the intersections of
infinite lines may be conceived of as made at a point which has no
dimensions, and as to thickness, if such a term can be employed, is
equal to the thickness of one single line.

HOW WE CONCLUDE THAT THE SURFACE TERMINATES

IN A POINT

An angular surface becomes reduced to a point when it reaches its
angle; or if the sides of this angle are produced in a straight line, then
beyond this angle there is formed another surface, less or equal or
greater than the first. Windsor mss. r 47

Every point is the head of an infinite number of lines, which com-
bine to form a base, and suddenly from the said base by the same lines
converge to a pyramid showing both its colour and its form.

No sooner is the form created or compounded than suddenly of itself
it produces infinite angles and lines, which lines spreading themselves
in intersection through the air give rise to an infinite number of angles
opposite to one another. With each of these opposite angles, given a
base, will be formed a triangle alike in form and proportion to the
greater angle; and if the base goes twice into each of the two lines of
the pyramid it will be the same with the lesser triangle.

Windsor mss. r 62

Archimedes has given the square of a polygonal figure, but not of
the circle. Therefore Archimedes never found the square of any figure
with curved sives; but I have obtained the square of the circle minus

642 MATHEMATICS

the smallest possible portion that the intellect can conceive, that is, the
smallest point visible. Windsor: Drawings 12280 v.

If into a vessel that is filled with wine as much water is made to en-
ter as equals the amount of the wine and water which runs out of it,
the said vessel can never be altogether deprived of wine. This follows
from the fact that the wine being a continuous quantity is divisible to
infinity, and therefore if in a certain space of time a particular quantity
has poured away, in another equal space of time half the quantity will
have poured away, and in yet another a fourth of the quantity; and
what is left is constantly being replenished with water; and thus always
during each successive space of time the half of what remains will be
poured out. Consequently, as it is capable of being divided to infinity,
the continuous quantity of the aforesaid wane will be divided during
an infinite number of spaces of time; and because the infinite has no
end in time there will be no end to the number of occasions on which
the wine is divided. Leic. 26 v.

Instrumental or mechanical science is the noblest and above all others
the most useful, seeing that by means of it all animated bodies which
have movement perform all their actions; and the origin of these move-
ments is at the centre of their gravity, which is placed in the middle
with unequal weights at the sides of it, and it has scarcity or abun-
dance of muscles, and also the action of a lever and counter-iever.

Sul Volo 3 r.