Harriot, Thomas. Mss. 6786

Harriot, Thomas, Mss. 6786

Bibliographic information

Author:	Harriot, Thomas
Title:	Mss. 6786

Permanent URL

Document ID:	MPIWG:2AX3C8P4
Permanent URL:	http://echo.mpiwg-berlin.mpg.de/MPIWG:2AX3C8P4

Copyright information

Original:	British Library
Digital-image:	British Library
Text:	Stedall, Jacqueline
Copyright for original:	British Library
Copyright for digital-image:	British Library
License for digital-image:
Copyright for text:	Max Planck Institute for the History of Science, Library
License for text:	CC-BY-SA

Table of contents

1. Euclides opticks Appollo: lib. 1. [tr: Apollonius, Book I ] Atomi Lib, 10: prop. 37. [tr: Book X, Proposition 37. ] Definitiones decimi Euclidis explicates per exempla numerorum [tr: Definitions from the tenth book of Eculid, shown by numerical examples. ] Clavius). 10. lib. Euclid. propositionum Annotationes in rationes scenographices Federico Comandino, post ptolemae planisphaerum [tr: Annotations on the scenographic ratios of Federico Comandino, in ] Definitionies Quinti libri [tr: Definitions in the fifth book ] 1) Pappus pag. 47 bb2 De numeris triangularibus. Mercators map In ptolomaicam propositione [tr: On a proposition of Ptolemy ] In prop. ptolomaica initio Finkij [tr: On a proposition of Ptolemy from the beginning of Finck ] In Cap. 17. Resp. pag. 29. Vieta [tr: From Chapter 17, Responsorum, page 29, Viète ] Ratio æqualitatis in infinitum. [tr: A ratio of equality at infinity. ] Ex bin: med: 4. [tr: From a fourth bimedial. ] Ex bin: med: 5. [tr: From fifth bimedials. ] as2) Page: 0

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S. JS. Harriman.
Mr Huss.
Mr Alisbury.
Mr Thorperly.
Mr

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Euclides

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[Note:

This page refers to Propositions I.36 and I.37 of Book I of Conicorum libri quattuor
I.36. If some straight line, meeting the transverse side of the figure touches an hyperbola or ellipse or circumference of a circle, and if a straight line is dropped from the point of contact as an ordinate to the diameter, then as the straight line cut off by the tangent from the end of the transverse side is to the straight line cut off by the tangent from the other end of that side, so will the straight line cut off by the ordinate from the end of the side be to the straight line cut off by the ordinate from the other end of the side in such a way that the corresponding straight lines are continuous; and another straight line will not fall into the space between the tangent and the section of the
I.37 If a straight line touching an hyperbola or ellipse or circumference of a circle meets the diameter, and from the point of contact to the diameter a straight line is dropped as ordinate, then the straight line cut off by the ordinate from the centre of the section with the straight line cut off by the tangent from the centre of the section will contain an area equal to the square on the radius of the section, and with the straight line between the ordinate and the tangent will contain an area having the ratio to the square on the ordinate which the transverse has to the upright.

Appollo: lib.
[tr: Apollonius, Book I]

37.p
36.
[tr: Propositions 36, 37; diameter:]

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[Note: A calculation of the number of atoms in 1 foot, 1 pase (= 5 feet), and 1 mile (= 1000 pases).]

1000 = \frac{1}{10}

10,000.
50,000.
50,000,000.

The number of miles in the circumference of the 60,000. miles circ.
The number of atoms in the circumference of the 3,000,000,000,000. atocirc. ter
The number of atoms in the diameter of the 1,000,000,000,000.
The number of atoms in 2,000 earth diameters, the supposed distance to the 2,000. diametri ad

The number of atoms from the earth to the ato. ad

The number of atoms in 2000 times the distance to the sun, the supposed distance to the fixed ato. ad [¿]stellæ[?]
Approximately double the previous line, that is, the number of atoms in the diameter of the sphere of the fixed maior
The cube of the previous line, that is, a rough and ready calculation of the number of atoms in the cubus maior

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[Note:

Calculations relating to formula (3) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f.

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[Note:

This page refers to Euclid X.37. In modern editions the relevant proposition is X.36, but Harriot's numbering matches that of both Commandino and Clavius.
If two ratioinal straight lines commensurable in square only be added together, the whole is irrational; and let it be called binomial.
Harriot's example is

2 + \sqrt{5}

There is also a reference below the diagram to Euclid VI.1:
Triangles and parallelograms which are under the same height are to one another as their

Lib, 10: prop.
[tr: Book X, Proposition 37.]

Si duae rationales potentia solum commensurabiles componantur,
tota irrationalis erit, nocetur autem ex binis
[tr: If two quantities commensurable in power only are combined, the whole will be irrational, moreover separated into two ]

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Definitiones decimi Euclidis
explicates per exempla
[tr: Definitions from the tenth book of Eculid, shown by numerical examples.]

1.) Magnitudeines (sive numeri)
[tr: Commensurable magnitudes (or numbers)]

2.)
[tr: Incommensurables]

3.) Commensurabiles
[tr: Commensurables in power]

4.)
[tr: Incommensurables]

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[Note: See also Add MS 6785, f. 210v. ]
de numeris
[tr: on perfect numbers]

Leida
Haga
Gouda
Leida
Ultrajestorum

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[Note:

This sheet contains an analysis of the first 40 propositions of Book X of Euclid's Elements, showing how each proposition depends on previous ones. The reference to Clavius is to his Euclidis Elementorum libri XV (1574, 1589, 1591, 1603,

Clavius). 10. lib. Euclid.

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If you call hym ffather which without respect of parson iudgeth
according to everie mans worke passe the time of your dwelling here in franc.
If you call hym ffather which without respect of parson iudgeth
according to everie mans worke passe the time of your dwelling
here in franc.
Banberynam

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ffather
But

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A man of wordes and not of deedes is like a garden full of weedes
A man of deedes and not of wordes is like a privifull of tourdes.
A man of

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Vuolffgangius [???], Per aliquot gentium migrationibus & lib. 12. fol.
Decades duæ devoniarum gentium emmigrationibus
in Illyrici occidentalis tractum.
Tonstall.
Lypsius de

Filtring paper.
4 & 8 & [???] glasses
2 glasses with elliptical
[???] for coyne.

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Annus
hoc
Ergo: Mensis

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[Note:

From right to left, the columns list:
units (1, 1, 1, 1,...)
lengths (1, 2, 3, 4,...)
triangular numbers (1, 3, 6, 10,...)
triangular-pyramidal numbers (1, 4, 10, 20,...)
triangular-pyramido-pyramidal numbers (1, 5, 15,

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[Note:

Harriot 19s letters refer to the diagram on page 3v of Ptolemaei planisphaerium

Annotationes in rationes scenographices Federico Comandino, post ptolemae planisphaerum
[tr: Annotations on the scenographic ratios of Federico Comandino, in]

pag.

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[Note:

This page appears to be related to page 177 of Stevin's L 19arithmetique pratique (1585). The problem that Stevin addresses there, as stated on page 174, is:
Estant donné ligne droicte, & deux nombres: Trouer vne ligne droicte en telle raison Ã la donnée, comme le nombre au nombre.
Stevin's examples are numerical, but Harriot is working in letters (as also in further notes on Stevin in Add MS 6782, f. 231).

Stevin.

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[Note:

The first half of a diagram continued on Add MS 6786, f.

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[Note:

The second half of the diagram from Add MS 6876, f.

A water worke found below
my Lord Chamberlayne at
his house in blacke friers
the 23 of January

564282v

The Lord
Chamberlaynes
Water Worke
Jan. 23.
1600

The plomber

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[Note:

The numersals 1 to 9 written with dots, and with various other symbols invented by Harriot for the

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571286

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575288

[Note:

A calculation of

\sqrt{3.458764513 . . .}

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581291

[Note:

Combinations of quantities generated by multiplication.
The letters

p

,

m

,

f

,

s

,

a

stand pondus magnitudo figura situs altitudo (altitude) (see also Add MS 6782, f. 181 and f. 48v).

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[Note:

Here Harriot appears to be devising new words for a counting system, with systematic endings for each new power of ten. In the first paragraph the basic numerals are: en, ben, tren, quen, quin, sinus, spinus, ..., which bear some resemblance to number words in languages derived from Latin, but in the second paragraph Harriot has adopted an alphabetical system: hon, an, ban, can, dan,

en ogen otem otille. emes emies illies
ben Begen Beten Betillia
tren trigen tritem tritilla
quen quagen quatem quatiliia
quinus quigen quitem quitillia
sinus sigen sitem sigillia
spinus spigen spitem

homilantu hoceman holeman honan holemil hobemil homil hocen hoben hon
amilantu aceman abeman anan acamil abemil amill acen aben an
cabeman banan bucemil cabemil bumilli bacen baben ban
cacen caben can
dacen daben dan
ecen eben en
fecen feben fen
gecen geben

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STevin problemata
Pappus
Apollonius

633317

[Note:

This page lists and compares the definitions in Book V of Euclid's Elements, as given by:
Federico Commandino, Euclidis Elementorum libri XV (1572),
Christophor Clavius, Euclidis Elementorum libri XV (1574, 1589, 1591, 1603, 1607),
and unnamed Greek

Definitionies Quinti
[tr: Definitions in the fifth book]

Græces
[tr: Earlier Greeks]

634317v

[Note:

A rough attempt at an analysis of teh contents and structure of Elements, Book I. For a more polished version see Add MS 6785, f. 337.

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[Note:

The third line is very nearly an anagram of the first; neither makes much sense in its own right. For frequency analyses of the letters see Add MS 6788, f. 251v and Add MS 6789, f, 456.

Sum mutatus amati mire præ sole lumen abigui
musa mutata mutis.
Sumamus amemus præbet illum origine

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[Note:

A double-page calculation of

\sqrt{(2 + \sqrt{2}}

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[Note:

A calculation of

\sqrt{(2 + \sqrt{2}}

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[Note:

The reduction shows that 1101101 represents 64 + 32 + 8 + 4 + 1 = 109.
Conversely, breaking down 109 into 64 + 32 + 8 + 4 + 1 shows that its binary representation is

Reductio.

[tr: Reduction.]

Conversio

[tr: Converse]

693347

[Note:

Examples of binary subtraction, addition, and multiplication, the latter done both directly and by successive addition.

Additionis Subductionis
[tr: An example of addition subtraction]

Additionis
[tr: An example of addition.]

[tr: Multiplication.]

Aliter, cum additionem

[tr: Another way, as successive addition.]

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[Note:

A list of numbers, names, and

202. Iantgennius Hessiæ. No. 11
245. [???] 12. No.
249.
250. 12. oct.
288. Cornelius Gemma. 14. N.
306. Ræslin.
320. Hagesius. 10. N.
369. Scultetus.
425. Nolthinus.
438. Winklerus.
448. alij

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Gordianus Nodus.
Labyrinthus Dædali.
Clava Herculis. Democritus his reason pro atomis.
Achilles. [???] reason.
Cerberus.
Tenebræ Farturea.
Cimmeriæ tenebræ.
Tentyritæ crcodilos non

The problem of a rectilinear rumbe
infinitely turning about the center
and yet the line finite to be moved
in an hour or any other given time to
be made
double to
passe to two

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[Note:

Diagrams representing various combinations of points (.), lines (-), and circles

[tr: better]

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[Note:

A plan for supplying water to a number of areas and buildings, including one that appears to be for Harriot's personal use.

cestern in the bathing room
cellar
cellar
privye
laundry
privy T. H.
The Stables
a cesperall for emptying the pipes
the plane
The farme
S. F. Darcy of Think
The Spring

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[Note:

This large folded sheet (f. 375v–376) is one of the few papers that Harriot dated: 12 December 1591. It is concerned with the piling of bullets, for which two questions naturally arise: 1. Given the number of bullets and the required shape of the base, how many bullets must be placed at each level, and how many levels? 2. Given a pile of a certain shape, how many bullets does it
(a) Square and oblong numbers:
The numbers 1, 2, 3, 4, ... down the key diagonal indicate the length of the shorter side of the base, and therefore also the height of the pyramid.
The numbers 1, 2, 3, 4, ... along the top indicate the length of the longer side of the base.
Thus for an oblong pyramid on a 3 x 7 base, move right from 3 in the key diagonal and down from 7 on the top line. The entries in the appropriate square show that there are 21 bullets in the base and 38 in the pyramid as a whole.
In particular, the square pyramidal numbers 1, 5, 14, 30, 55, 91 ... appear in the squares immediately to the right of the key diagonal; thus a square pyramid with base 3 x 3 has 9 bullets in its base and 14 bullets in
(b) Triangular numbers:
The triangular pyramidal numbers 1, 4, 10, 20, 35, ... appear as the lower entries in the squares below and to the left of the key diagonal, to which they are joined by short lines. Thus a triangular pyramid with three levels contains 6 bullets in its base and 10 bullets in total. As Harriot points out in his note on f. 376, the triangular numbers 1, 3, 6, 10, 15, ... may be read off as the differences of the oblong numbers in the previous row. Conversely, the entire table can be generated by successive addition from the triangular and square

These are those speciall
groundplats upon the which
may be orderly piled bullets:
The triangle: the square: and
the

Concerning piling there are two
questions: one which the nomber
of bullets to be piled being given
with the forme of the groundplat,
to know how many must be placed
in every rounde, with how many
roundes in the sayd

751376

The second a pile being made to knowe
the number of bullets therein conteyned.
For the awnswering of which two questions
this table of some set downe calculations for for the purpose. December. 12.

The forme of the progression of the triangle row
is the excesse of the progression of the of the
square oblong plats next adjoyning
as. 10 of 30, 40, 50 &c.
or. 15 of 55, 70, 85, &
Thereby the squares & triangles being knowne
all any of the oblongs may be knowne by

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753377

[Note:

Arrangements of dots as singletons or pairs in rows of four.
Perhaps a counting system in base

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[Note:

Arrangements of dots in pattersn similar to those on the previous folio (Add MS 6786, f. 377), now also in conjunction with signs of the zodiac.

756378v

[Note:

Arrangements of dots in patterns similar to those on the previous folio (Add MS 6786, f. 377), now arranged in a circle.

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[Note:

Some examples of Stevin's

777389

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[Note:

A set of Napier's bones, and some

788394v

[Note:

Calculations using Napier's bones, continued from overleaf Add MS 6786, f.

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One chain is 4 poles. One pole is 5.5 yards, or 198 792 inches, 4 poles; the length of the
7920 tenths of an inch; 4 poles: or length of the
792 tenths or:

79 \frac{2}{10}

inches or 6 foot 7 inches

\frac{2}{10}

of inches; is
for one tenth parte of the
one of those partes must be divided into 10
other

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[Empty page]

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801401

[Note:

This page referes to Stevin's L 19arithmétique … aussi l 19algebre, page 306. There Stevin solves the equation 1(3) = 6(1) + 40 (in modern notation

x^{3} = 6 x + 40

), leading to the cube roots writen down by Harriot at the top of the page.

√C.20 + √392 + √C.20 – √392. Stevin pag.

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809405

To set the elements of the doctrine of triangles
as deviding of arke by the sines in
datum rationum. & doubling.
The elements by versed sines
ut lat. ad lat. ita. signs aut. ad v. ang.
& the prosthephaeresis.
in

To set after the doctrine in species.
& in the forme of æquations, whereby
the rules may be [???]:
The compound numbers only to be brought
into one

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[Empty page]

815408

[Note:

The reference to Pappus is to Commandino's edition of Books III to Mathematicae collecitones (1558). The proposition on page 47 is Proposition IV.14.
Theorema XIIII. Propositio XIIII.
Sint duo semicirculi BGC BED: & ipsos contingat circulus EFGH: a cuius centro A ad BC basim semicirculorum perpendicularis ducatur AM. Dico ut BM as eam, quæ ex centro circuli EFGH, ita esse in prima figura vtramque simul CB BD ad earum excessum CD; in secunda vero, & tertia figura, ita esse excessum CB BD ad vtramque ipsarum CB
Let there be two semicircles BGC and BED, and their touching circle EFGH, from whose centre A to BC, the base of the semicircle, there is drawn the perpendicular AM. I say that as BM is to that line from the centre of the circle EFGH, inthe first figure will be CB and BD togher to their excess, CD; but in the second and third figure, it will be as the excess of CB over BD to both of CB and BD
For Harriot's diagrams for this proposition, see Add MS 6784, f. 203 and f. 204; this page shows only calculations of ratios.

1) Pappus pag.

816408v

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819410

[Note:

Rough working, similar to page 14 of the 'Magisteria' (Add MS 6782, f.

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829415 Work on eliminating the linear term from an equation of degree

bb2

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833417

[Note:

The numbers in the table at the top of the page are, from left to right: units, lengths, triangular numbers, triangular-pyramidal numbers, and so on.
The table in the centre of the page repeats the first and third columns from the table at the top, but now the triangular numbers are written in multiplicative form, thus,

6 = \frac{3 \times 4}{1 \times 2}

.
The third table gives all the rows in multiplicative form, thus in the seventh

84 = \frac{7 \times 8 \times 9}{1 \times 2 \times 3}

.
The final row gives general formulae for the

n

th row.
See also page 1 of the 'Magisteria' (Add MS 6782, f. 108), for which this is possibly a

De numeris

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[Empty page]

840420v

[Note:

Calulation of the number of days and hours in (i) 235 lunar months; (ii) 19 solar years; (iii) Julian years, and the differences between

235.
[tr: 235 lunar months.]

19.
[tr: 19 solar years.]

19.
[tr: 19 Julian years.]

841421

[Note:

Further calculations relating lunar months to solar

842421v

[Empty page]

843422

Mercators

Ptol:
[tr: Prolemy's geography]

Hierosolyma.
[tr: Longitude of Jersualem]

Alexandria.
Babilon.
Roma.
Londinium.

Huena
ab

Huena ab

Huena
ab

Huena a

Huena
ab

Tycho habet 1.35. diff. long
Huenæ ab
[tr: Tycho has 1.35. as the difference in longitude betweeh Huena and Alexandria.]

844422v

[Empty page]

845423

[Note:

Calculations of

\sqrt{7.8696044 . . . = \sqrt{(π)^{2} - 2}}

and of

\sqrt{5.8696044 . . . = \sqrt{(π)^{2} - 4}}

846423v

[Note:

Calculations of

\sqrt{5.9348022 . . . = \sqrt{\frac{(π)^{2} + 2}{2}}}

847424

[Note:

Note that

9.8696044 = (π)^{2}

848424v

[Note:

Calculations of

\sqrt{1.02192 . . .}

and of

\sqrt{25.93527 . . .}

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[Empty page]

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[Empty page]

857429

These wordes are written
with China inke.
so likewise are these:
But these are written with London

With China inke these wordes.
With China inke these wordes.
were written with thick
inke &

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[Empty page]

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[Empty page]

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[Empty page]

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[Empty page]

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[Empty page]

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[Empty page]

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[Empty page]

879440

[Note:

This sheet contains Harriot's exploration of Proposition 2, Lemma 4, from Sphaerica Menelai of Menelaus, as translated by Maurolico (1558). The text includes a supplement by Thabit ibn Qurra (see Add MS 6787, f. 306v to f. 308 and f. 309v).
For Harriot's copy of Propositions 1 and 2 see Add MS 6787, f. 306 to f. 309v. The pages relevant to Proposition 2, Lemma 4, are f. 307, f. 307v, and f. 308.

Si fuerint sex quantitates, quorum ratio primæ ad secundum: componatur ex rationibus,
tertia ad quartam, et quintæ ad sextam. Tum solidum sub prima quarta et sexta
contentum, æquale erit solido sub secundo tertia et quinta
[tr: If there are six quantities, where the ratio of the first to the second is composed of the ratio of the third to the fourth and of the fifth to the sixth, then the solid content of the first, fourth, and sixth will be equal to the solid composed of the second, third, and ]

Sex
[tr: The six quantities]

Ergo solidum

a d z = b g e

[tr: Therefore the solid

a d z = b g e

]

Varia transpositio utrinque solilidi solidi
[tr: Various transpositions of the terms of both solids.]

Habitudines æqualium solidorum.
[tr: Forms of equal solids]

Thebit ben Corat.
et Mau
[tr: Thabit ibn Qurra and Maurolico.]

[tr: compositions]

componentes rationes
[tr: components of the transposed ratios]

Si consequentes fiant antecedentes
et antecedentes consequentes; sunt alia
36
[tr: If the consequents become antecedents and antecedents consequents, there are another 36 compositions.]

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881441

In ptolomaicam
[tr: On a proposition of Ptolemy]

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[Empty page]

883442

Data
[tr: given magnitude]

Data
[tr: Gien ratio]

Quæruntur
[tr: Magnitudes sought]

vel cuiuslibet
[tr: or any others one wishes]

884442v

[Empty page]

885443

[Empty page]

886443v

[Note:

For the proposition referred to here, see Add MS 6785, f. 179. See also Add MS 6786, f. 441 to f. 444.

In prop. ptolomaica initio
[tr: On a proposition of Ptolemy from the beginning of Finck]

887444

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[Empty page]

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[Empty page]

893447

[Empty page]

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[Empty page]

895448

Ex Stigelij lib. de circulis
calistibus. et Doctrina

896448v

[Empty page]

897449

[Empty page]

898449v

[Empty page]

899450

[Note:

One of several pages containing a mnemonic for the digits of

π

, by making

1 = a

, 2 = b

,

3 = c

, a n d s o o n .

cadaeibf

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901451

[Empty page]

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[Empty page]

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[Empty page]

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[Empty page]

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[Empty page]

906453v

[Note:

The reference on this page is to Variorum responsorum liber VIII, Chapter
Theorema.
Si fuerint magnitudines continue proportionales: erit ut terminus rationis major ad terminum rationis minorem, ita differentia compositæ ex ombinbus & minimæ ad differentiam compositæ ex omnibus &
If there are magnitudes in continued proportion, then the ratio of the greatest term to the least will be as the difference between the sum of all and the minimum to the difference between the sum of all and the maximum.
Harriot's notation here is the same as

In Cap. 17. Resp. pag. Vieta
[tr: From Chapter 17, Responsorum, page 29, Viète]

Si fuerint magnitudines continue proportionales: Erit ut terminus
rationis major ad terminum rationis minorem; ita differentia
compositæ ex ombinbus et minimæ ad differentiam compositæ ex
omnibus et
[tr: If there are magnitudes in continued proportion, then the ratio of the greatest term to the least will be as the difference between the sum of all and the minimum to the difference between the sum of all and the ]

Sit maxima.

D

.
minima.

X

.
composita ex omnibus.

F

ratio Maioris ad minore.

D

,

B

[tr: Let the greatest be

D

,
the least

X

,
the sum of all

F

,
the ratio of a greater to the lesser

D : B

]

Alius sic: si fuerint magnitudes continue proportionales:
ut prima ad secundam: ita omnes antecedentes ad omnes
[tr: Otherwise thusL if there are magnitudes in continued proportion, as the first is to the second, so are all the antecedents to all the ]

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12£ sterling. = 1£ gold

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[Note:

Powers of

(b + c)

up to

(b + c)^{5}

. Each power is calculated from the previous one by multiplication.
Note the use of cossist

r

for a first power,

z

for a square,

c

for a cube,

z z

for a square-suare or fourth

ß

for a sursolid or fifth power.
Below the main table is a list of the final sums, including the sixth power (

z c

), which has not been calculated on this page but which can be deduced from the pattern for the previous cases.
For a similar table see Add MS 6782, f. 276.
This table underpins the method of root extraction taught by Francois Viete in De numerosa potestatum resolutione

species figuratorum
ex binomia
[tr: terms of figurate numbers from binomial roots]

(sunt canones pro
extractione
[tr: (these are the canons for the extraction of roots.)]

(Demptis numeris sunt species
continue proportionalium,
in minimis terminis
ut

b

, ad

c

[tr: (Without the numbers, the terms are in continual proportion, in the ratio expressed in lowest terms as

b

to

c

]

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vale vale cane ne titules mandataque

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Ratio æqualitatis in
[tr: A ratio of equality at infinity.]

A cipher being the last of this lower progression what wilbe
over it, of the hyer, proceeding & continuing, as it is

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William Yates William Yates. Christopher

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Ex bin: med:
[tr: From a fourth bimedial.]

bin.
[tr: A sixth binome.]

bin. 5 ex bin: med.
[tr: A fifth binome from a first bimedial.]

Ex bin: med:
[tr: From fifth bimedials.]

bin. 5 ex bin: med.
[tr: A fifth binome from a first bimedial.]

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[Note:

The author of this fragment is unkown. However, the same form of address was used by Harriot's friend Nathaniel Trporley, see Add MS 67888, f. 117.

TO HIS VERY LOVING FRENDE
MASTER THOMAS HARRIOT
GEVE THESE AT

981491

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1016508v See also Add MS 6783, f. 174 (d.10) where the same quadratic equation is solved for

d

as2

ad expurgandam

a a

[tr: for the removal of

a a

]

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[Note:

Examples of arithmetic, geometric, and harmonic porgressions, and rules for their general terms. (The terms of a harmonic progression are the reciprocals of an arithmetic porgression.)

Arith. 3. 6.
Geo. 3. 6.
Har. 6. 3.

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[Note:

This page contains jottings on both binary and ternary arithmetic.
The calculation on the left shows 101101 (= 45) multiplied by 1011 (= 11); the answer, 111,101,111 (= 495), is then divided by 1011 (= 11) to recover 101101 (= 45).

ternaria

[tr: ternary]

binaria

[tr: binary]

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[Note:

The beginning of a page 34.3 for the 'Magisteria', not used in the final version which has only pages 34.1 and 34.2 (see Add MS 6782, f. 141 and f. 142).

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red sage. 624.
une. 1073.
cummin. 908.
cammomill. 616.
otes. 69.
wall une.

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poles
London
Syon
London Durham house
The East

1109555

Brought by Mr Alisbury from Bristow
Avacardus
The kernel
good to eat.
The ioyce of the vine
good for tetters.
quod de Loan.
The Apot:
in

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[Note:

Pythagoreadn triples with hypotenuse 65, 85, 221, and

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