# 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 [26] 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 [9] 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 [10] 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 [12] 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 [30] v. and 79 [31] 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 [32] 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.