Harriot, Thomas, Mss. 6784

Bibliographic information

Author: Harriot, Thomas
Title: Mss. 6784

Permanent URL

Document ID: MPIWG:21SW4DPY
Permanent URL: http://echo.mpiwg-berlin.mpg.de/MPIWG:21SW4DPY

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. De resectione rationis De resectione rationis AB) De resectione rationis De resectione rationis De resectione rationis De resectione rationis De resectione rationis AB) De resectione rationis De resectione rationis 2.AB) De resectione rationis AC) De resectione rationis AC.1) De resectione rationis De resectione rationis 2.BC) De resectione rationis 1.BC) De resectione rationis De resectione rationis Pappus 171. ad resectione rationis De resectione rationis De resectione spatij, problema a) Poristike De sectione rationis b.1) De sectione rationis b.2) De sectione rationis b.3) De sectione rationis b.4) Lemma ad sectionem rationis et spatij De resectione rationis Diagrammata Snellij [tr: Snell's diagrams ] Lemma. 1. Appol. Bat. pag. 81. Aliter de 12. 2i Euclidis et 13. [tr: Another way for Euclid II.12 and 13. ] phys. lib.6. Cap. 1 [tr: Physics, Book 6, Chapter 1 ] Arist. lib. 6. Cap. 2 [tr: Aristotle, Book 6, Chapter 2 ] Residuum 5a operationis, G. [tr: The rest of the working (5) on G. ] 5a operatio, G. [tr: Working (5) on G ] Operatio. D. [tr: Working on D ] Residuum 3a operationis, G. [tr: The rest of the working (3) on G. ] 3a operatio. G. [tr: Working (3) on G ] β.11 De tactibus β.2 β.3 β.4 β.5) β.5.2o) β.6.) α.1 α.3 α.2 β.1 De tactibus 7. (o o) De tactibus Probl. 6 (. o -) 6) De tactibus [tr: On touching ] 6.) 7.) Exegesis arithmetica pro ph radio. [tr: Arithmetical exegesis, for radius ph. ] 6.) Arithmetica Exegesis radij by [tr: Arithmetical exegesis, for radius by. ] Arithmetica exegesis radij ah cæteris datis. [tr: Arithmetical exegesis, for radius ah, given the rest. ] Geometria exegesis ipsius radii ah. [tr: Geometric exegesis, for the same radius ah. ] 5.) pappus. prop. 13. pag. 49. 4.) pappus. pag. 47. [tr: Pappus, page 47. ] pappus. pag. 47. [tr: Pappus, page 47. ] 2) pappus. pag. 47 3) pappus. pag. 47 Operatio. G. [tr: Working on G ] Residuum operationis. G. 3a notatio triangularium per notas generales. [tr: 3rd notation for triangular numbers, in general symbols. ] Examinatio æquationis per numeros [tr: An examination of an equation in numbers ] 3. 4. 5. 1) Operationes logisticæ, in notis [tr: The operations of arithmetic in symbols. ] 2) 3) 4) 5. Appolonius. pag. 5. 6. [tr: Apollonius, pages 5, 6. ] Ad appolonium. pa. 5. 6. [tr: On Apollonius, pages 5, 6 ] prop. 18. Supplementi. [tr: Proposition 18 from the Supplementum ] prop. 18. Supplementi. [tr: Proposition 18 from the Supplementum ] prop. 17. Supplementi. [tr: Proposition 17 from the Supplementum ] prop. 16. Supplementi. [tr: Proposition 16 from the Supplementum ] prop. 15. Supplementi [tr: Proposition 15 from the Supplementum ] Hinc tale Consectarium potest efferri [tr: Here a Consequence of this kind may be inferred ] prop. 12. Supplementi [tr: Proposition 12 from the Supplementum ] Consectarium [tr: Consequence ] prop. 10. Supplementi [tr: Proposition 10 from the Supplementum ] prop. 11. [tr: Proposition 11 ] Consectarium [tr: Consequence ] Ad Corollorium prop. 7. Supplementi. Et ad cap. 5. Resp. lib. 8. pag. 4. [tr: On a corollary to Proposition 7 of the Supplement. Also Chapter 5, Variorum liber responsorum, page 4. ] prop. 7. Supplementi de corrollario [tr: Proposition 7 of the Supplement, on a corollary ] In Cap. 5. Resp. lib. 8. pag. 4. [tr: Chapter 5, Variorum liber responsorum, page 4. ] prop. 7. Supplementi [tr: Proposition 7 of the Supplement ] a) Achilles b) Achilles 1.) De progressione geometrica. [tr: On geometric porgressions ] 2.) De progressione geometrica. [tr: On geometric porgressions ] 3.) De progressione geometrica. (ut Vieta in var: resp.) [tr: On geometric progressions (as Viete in Variorum responsorum) ] De progressionibus. finitis & infinitis. [tr: On finite and infinite progressions ] propositiones 2i Euclidis [tr: Propositions from the second book of Euclid ] 1.) De reductione æquationum [tr: On the reduction of equations ] 3.) 1)B) De reductione æquationum [tr: On the reduction of equations ] 1o. de bisectione. [tr: 1. on bisection ] 2o. De sectione in tres partes. [tr: 2. On sectioning into three parts. ] Ad generationes sequentium specierum æquationum [tr: On the generation of the following types of equation. ] De infinitis. Ex ratione motus, temporis et spatij. [tr: On infinity. From the ratio of motion, time and space. ] Page: 0
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De resectione
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Pappus 171. ad resectione
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De resectione
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De resectione spatij,
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De sectione
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Lemma ad sectionem rationis
et
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[Note:
De
[tr: On infinity]


Maior et Maior rationum infinitum.
fit termini minores et minores; cum probuerit indivisibilibis
ratio tandem
[tr: A greater and greater infinite ratio. the terms are smaller and smaller; while from indivisibles there will eventually come an infinite ratio.]
HA ad IA non potest
esse maior BA ad BC.
terminis scilicet
[tr: HA to IA cannot be greater than BA to BC. the terms of course ]
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[Note:
The reference on this page is to Willebrord Apollonius Batavus


Diagrammata

[tr: Snell's diagrams]
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Calculation of (b+c+d)(b+c-d)(b-c+d)(-b+c+d)


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Calculation of (b+c+d+f+g)(b+c+d+f-g)(b+c+d-f+g)


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Graecia
prævenians.
excitans.
vocans.
operans.
provens.
comians.
cooperans.
adiunans.
concomitans.
subsequens.
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[Note:
The references on this page are to Pappus, Book 7, and to Giambattista Benedetti,Diversarum speculationum mathematicarum et physicarum liber


sit triangulum acd
[tr: let there be a triangle acd]

dico
[tr: I say that]
sit ae perpendicularis ad, cd
[tr: let ae be perpendicular to ad]
Unde
[tr: whence it follows]
Vide, Pappum. lib. 7. prop: 122. pag. 235.
et: Jo: Baptistum Benedictum pag.
[tr: See Pappus, Book 7, Proposition 122, page 235; and Johan Baptista Benedictus, page 362]

[tr: turn over]
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Lemma. 1. Appol. Bat. pag.
Sit:
Dico quod:
nam in utraque
[tr: Let:
I say that:
for in the both ]
Sed ita
[tr: But it is thus in Snell.]
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[Note:
This page contains symbolic versions of Euclid Book II, Propositions 12 and 13:
II.12.In obtuse-angle triangles the square on the side opposite the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle.
II.13. In acute-angled triangles the square on the side opposite the acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acute


Aliter de 12. 2i Euclidis
et
[tr: Another way for Euclid II.12 and 13.]
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Calculation of (b+c-d)(b-c+d)(-b+c+d)


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phys. lib.6. Cap.
[tr: Physics, Book 6, Chapter 1]
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Arist. lib. 6. Cap.
[tr: Aristotle, Book 6, Chapter 2]
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[Note:
Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f.


Residuum 5a operationis, G
[tr: The rest of the working (5) on G]
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[Note:
Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f.


5a operatio, G
[tr: Working (5) on G]
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Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f.


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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.


=d. (si N=0.)
vel, cuivis d
[tr: or, for any d]
=c.
[tr: any]
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Calculations relating to formula (3) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f.


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Operatio. D
[tr: Working on D]
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Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f.


Residuum 3a operationis, G
[tr: The rest of the working (3) on G]
Residuum 4a operationis, G
[tr: The rest of the working (4) on G]
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[Note:
Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f.


3a operatio. G
[tr: Working (3) on G]
4a operatio
[tr: Working (4) on G]
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β.11 De

[tr: beware]
d. est centrum circuli
circumscribentis.
Tria traingula. abc. abd. bcd.
habet periferias
[tr: d is the centre of the circumscribing circle.
The three triangles, abc, abd, bcd have equal ]
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β
Δ,abd, latera

[...]

cuius superficies ut
[tr: Triangle abd, with sides:

[...]

whose surface is as ]
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β
Δ,abc, latera

[...]

cuius superficies ut
[tr: Triangle abc, with sides:

[...]

whose surface is as ]
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β
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[Note:
The reference in the top right hand corner is to Apollonius Gallus (1600), Problem
Problema IX.
Datis duobus circulis, & puncto, per datum punctum circulum describere quem duo dati circuli contingat.
IX. Given two circles and a point, through the given point describe a circle that touches the two given


β
Vide: Appol. Gall. prob.
[tr: See Apollonius Gallus, Problem IX.]
Aberratur de modo contingendi
circulos posititios alias operatio bona
vide igitur β.5.2o
[tr: There is an error in the method of touching the supposed circles, othersie the working is good; therefore see shee β]
[Note: Sheet β.5.2 is Add MS 6784, f. 181. ]
radius circuli posititij (b) minoris
posititij (c) maioris
distantia
[tr: radius of the smaller supposed circle, b,
of the greater supposed circle, c,
distance of the ]
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[Note:
A continuation of the work on Add MS 6784, f.


β.5.2o
Vide: Appol: Gall. prob. 9.
fig:
[tr: See Apollonius Gallus, Problem IX, figure 2.]
radius circuli posititij (b) minoris
posititij (c) maioris
distantia
[tr: radius of the smaller supposed circle, b,
of the greater supposed circle, c,
distance of the ]
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β
radius circuli posititij (b)
posititij (c)
distantia
[tr: radius of the supposed circle, b,
of the supposed circle, c,
distance of the ]
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α
data

[...]

Quæritur:
[tr: given

[...]

Sought, ]
Δ,abd, latera

[...]

cuius superficies ut
[tr: Triangle abd, with sides:

[...]

whose surface is as ]
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α
Δ,acd, latera

[...]

cuius superficies ut
[tr: Triangle acd, with sides:

[...]

whose surface is as ]
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α
Δ,bcd, latera

[...]

cuius superficies ut
[tr: Triangle bcd, with sides:

[...]

whose surface is as ]
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β.1 De
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7. (o
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De tactibus
Probl. 6 (. o
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6) De
[tr: On touching]
problema.
Datis tribus circulis
sese mutuo contingentibus:
invenire quartum circulum
qui mutus tangetur in
[tr: Problem.
Given three circles, mutually touching, to find a fourth circle that is mutually touched by those ]
Sint tres dati circuli brd,
rth, hue, sese mutuo contingentes
in punctis r, h, e. cuius centra z, p, a.
Agatur recta ap in continuum

[...]

Agatur fb recta contingens
circulum brd in puncto b.
Agatur recta bz in continuum quæ secabit
circulum cuius centrum z in d puncto.
fiat, uai recta, parallela bz.
Et ad lineam bz productam sint per-
 pendicularis uq, il
[tr: Let the three given circles be brd, rth, hue, mutually touching at the points r, h, e, whose centres are z, p, a.
There is constructed the extended line ap.

[...]

There is constructed the line fb touching the circle brd at the point b.
There is constructed the extended line bz which will cut the circule whose centre is z in the point d.
Let the line uaibe parallel to bz.
And to the extended line bz let there be perpendiculars uq and il]
Bissecetur bc in puncto y.
Centro y, intervallo yb,
describatur circulus.
Dico quod: ille est circulus quæsitus
et contingit tres datos
in puncto b, t, g
[tr: Let the line bc be bisected at the point y.
With centre y, radius yb, there is drawn a circle.
I say that this is the circle sought, and that it touches the tree given circles at the points b, t, g]
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6.)
Sint tres dati circuli, brd,
hue, bgc, sese mutuo
contingentes in punctis b, g, e,
cuius centra, z, a, y
[tr: Let there be three given circles, brd, hue, bgc, mutually touching in the points b, g, e, whose centres are at a, a, y]
Oportet invenire circulum
contingentem tres datos:
(nempe, rht, cius centrum, p
[tr: One must find the circle touching the three given ones (that is, rht, with centre p]
Per centra z y, agatur recta
et continuetur ad utraque partes
et fit b, z, y, d, c.
Et ad illam lineam bc, fit am
perpendicularis.
Continuetur ad partes contrarias
usque ad k, et fit sk=sa.
Tum primo, agatur recta bs
quæ secabit periferiam circuli
cuius centrum a in puncto h.
Secundo, agatur recta bk
quæ secabit ah productam in
puncto p.
Ultimo, centro p, intervallo
ph describatur circulus.
Dico quod: ille est circulus quæsitus
et contingit tres datos in
punctis, h, r, t
[tr: Through the centres z and y, a line is drawn and continued on both sides, and so there b, z, y, d, c.
And to that line bc, let am be perpendicular.
It is continued to both sides as far as k, and let sk=sa.
Then, first, there is drawn the line bs, which will cut the circumference of the circle with centre a in the point h.
Second, there is drawn the line bl, which will cut ah extended, in the point p.
Finally, with centre p and radius ph, there is drawn the required circle.
I say that this is the circle sought, and it touches the three given at the points h, r, t]
Exegesis arithmetica
pro ph
[tr: Arithmetical exegesis, for radius ph]
Datorum circulorum radii
dati sunt, et centrorum
distantiæ.
Ergo lateri trianguli zay
data sunt. Inde perpendicularis
am, et recta dm. Inde tota bm.
Inde datur ba. Inde bθ.
Tum cum datur sa et am, datur
sm et inde bs. Et cum datur
xb et bθ, datur bh et hs.
Tum lineæ bc fit bf ad angulos
rectos et ap pro-
 ducta concurret cum illa
in puncto f. fb et fh sunt
æquales. et triangulum fbh
simile est triangulo ash,
cuius latera data sunt. et
antea datum fuit bh. ergo dantur
fb et fh.

[...]

Ergo tota fa datur

[...]

Ergo datur ap
sed antea nota fuit ah,
ergo hp datur
Quod
[tr: The radii of the fiven circles are given, and the distances of their centres.
Therefore the sides of the triangles zay are given. Hence the perpendicular am, and the line dm. Hence the total, bm. Hence there is given ba. Hence bθ. Then since sa and am are given, sm is given and thence bs. And since xb and bθ are given, bh and hs are given.
Then the line bc is at right angles to bf and ap extended meets with it at the point f. The lines fb and fh are equal. And the triangle fbh is similar to triangle ash, whose sides are given. And earlier bh was given. Therefore fb and fh are given.

[...]

Therefore the total fa is given.

[...]

Therfore there is given ap, but earlier ah became known, therefore hp is given.
Which was ]
Per doctrinam sinuum
opus abbreviatur, sed
alia method ut
[tr: By the doctrine of sines, the work is shorter, but another method, as convenient.]
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399200
Arithmetica Exegesis
radij by
[tr: Arithmetical exegesis, for radius by]
Datorum circulorum radij dati
sunt, et centrorum distantiæ
Ergo lateri trianguli z, p, a,
cum sit, ut ah,hp:af,fp.
datur fp. et fh cui æqualis
fb
[tr: The radii of given circles are given, and the distances of their centres.
Therefore the sides of the triangle zpa, and since ah:hp=af:fp, there is given fp, and fh, which is equal to the angent fb]
Ex fb et bz datis, datur fz.
Sunt igitur duo triangula
datorum laterum fbz, fpz.
constituuntur super eandem
basim fz. datur igitur verti-
 cum distantia pb
[tr: From fb and bz, given, there is given fz.
Therefore there are two triangles with given sides fbz, fpz, constructed on the same base fz.
Therefore the vertical distance pb is ]
Ex triangulo bpz datorum laterum
datur zn et pn perpendicularis
nota igitur bn.
fiunt nη et nλ, æquales radio
circuli circa p.
Dantur, igitur bη et bλ.
Tum:
Datur igitur bc, cuius dimidium
by, radius
[tr: From the triangle bpz with given sides there is given zn, and the perpendicular pn is known, therefore bn.
There are constructed nη and nλ, equal to the radius of the circle about p.
Therefore there are given bη and bλ.
Then:
Therefore there is given bc, whose half, by, is the sought radius.
]
Per Canonem triangulorum
alia methodo ut covenit, operatio fit

[tr: By the Canons for triangles, there is another method, as convenient, which may be carried ore briefly.]
Nota.
per puncta η et λ
fit etiam geometrica
constructio, loco q, l
[tr: Note.
Through the points η and λ there may also be carried out a geometric construction, instead of q and l]
Arithmetica exegesis
radij ah
cæteris
[tr: Arithmetical exegesis, for radius ah, given the ]
Datorum circulorum radij dati
sunt, et centrorum distantiæ
Ergo lateri trianguli z, p, y,
Datur igitur perpendicularis
pn, et linea zn. Unde nota
fit bp.
Cum data pn et po
unde data bo.
Tum, trianguli bpo latera sunt
nota; unde nota perpendicularis
pu. Et linea ou, cui æqualis uh.
Dantur igitur oh et bh.
Dantur igitur hf et pf.
Denique fiat:
Datur igiture ah, quod

[tr: The radii of given circles are given, and the distances of their centres.
Therefore the sides of the triangle zpy.
Therefore there is given the perpendicular pn, and the line zn. Whence there is known bp.
Since pn and po are given, there is given bo.
Then the sides of triangle bpo are known, whence the perpendicular pu is known. And the line ou, which is equal to uh.
Therefore there are given oh and bh.
Thereofre there are given hf and pf.
Then let there be constructed:
Therefore there is given ah, which was ]
Geometria exegesis
ipsius radii ah
[tr: Geometric exegesis, for the same radius ah]
Trium datorum circulorum
centra z, p, y, connectantur.
per z, y fit bc acta
fb faciat angulos rectos cum bc.
Ita pn; quæ secabit circulum
circa p, in puncto o.
Agatur bo, quæ producta secabit
eandem circulum circa p, in h.
Agatur hp et producatur ad
utraque partes quæ secabit bf
in puncto f.
Tum fiat:
Datur igiture ha, et centrum circuli

[tr: Let the centres of the given circles, z, p, y, be connected.
Through z, y let bc be constructed; fb makes a right angle with bc.
Thus pn, which cuts the circle about p in the point o.
Let there be constructed bo, which extended sill cut the same circle about p at h.
Let hp be constructed and extended on both sides, which will cut bf in the point f.
Then:
Therefore there is given ha, and the centre of the circle ]
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[Note:
The reference to Pappus is to Commandino's edition of Books III to Mathematicae collecitones (1558). The proposition on page 48v–49 is Proposition IV.15 (not 13, as Harriot appears to have written). A diagram for this proposition appears on Add MS 6784, f. 202; this page shows only calculations of ratios.
Theorema XV. Propositio XV.
Iisdem positis describatur circulus HRT, qui & semicirculos iam dictos, & circulum LGH contingat in punctis HRT, atque a centris A P ad BC basim perpendiculares ducantur AM PN. Dico vt AM vna cum diametro circuli EGH ad diametrum ipsius, ita esse PN ad circuli HRT
The same being supposed [as in Proposition 14], there is drawn the circle HRT, which touches both the semicircles already given and the circle LGH, in the points H, R, T. And from the centres A and P to the base there are drawn perpendiculars AM and PN. I say that as AM together with the diameter of the circle EGH is to that that diameter itself, so is PN to the diamter of the circle


5.) pappus. prop. 13. pag.
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403202
Sint duo circuli bed et bgc
contingant se in puncto b.
sit recta per centra bopcd.
oportet describere circulum
contingentem duos circulos
datos, et lineam bc
[tr: Let there be two circles bed and bgc touching in the point b.
Let the line through the centre be bopcd.
One must draw the circle touching the two given circles and the line bc]

[...]

Jungantur puncta r, d.
fiat sk parallela rd.

[...]

[tr:
[...]

Let the points r, d be joined.
Let sk be parallel to rd.

[...]
]
Bisecetur kl, puncta m.
agatur ad angulos rectos, ma.
fiat, ma=mk.
agatur oa, quæ secabit periferi-
 am minoris circuli in e.
agatur pag, quæ secabit perife-
 riam maioris circulam in g.
Dico quod: am=ag=ae.
et ideo, circulus per m, g, e,
erit
[tr: Let kl be bisected at the point m.
There is constructed at right angles ma.
Let ma=mk.
There is constructed oa, which will cut the circumference of the smaller circle at e.
There is constructed pag, which will cut the circumference of the larger circle at g.
I say that am=ag=ae, and therfore the circle through m, g, e will be the one ]
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[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. Harriot's diagram is the same as the one given by Commandino except for his use of lower case letters. A second diagram for the same proposition appears on Add MS 6784, f. 204.
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


pappus. pag.

[tr: Pappus, page 47.]
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[Note:
A further diagram for Pappus, Mathematicae collectiones, Propostion IV.14. See also the previous folio, Add MS 6784, f. 203.


pappus. pag.

[tr: Pappus, page 47.]
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[Note:
Further work on Pappus, Propostion


2) pappus. pag.
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Further work on Pappus, Propostion


3) pappus. pag.
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[Note:
Lists of variations of increasing (c) and decreasing (d) columns, together with other rough work for the 'Magisteria' (Add MS 6782, f. 107 to f. 146v).
This page is important because it carries a date, day, time, and year: June 28 (Sunday) 10.30am,


De causa reflexionis ad angulos
[tr: On the cause of reflection at equal angles.]
June 28. .ho: 1012
ante mer:
[tr: June 28 (Sunday) 10.30am 1618]
414207v
[Note:
Further lists of variations of increasing (c) and decreasing (d) columns (see Add MS 6784, f.


415208
[Note:
Difference tables similar to those on pages 10 and 11 of the 'Magisteria' (Add MS 6782, f. 117 and f.


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[Note:
Formulae for entries in the d column of a difference table, similar to those on page 14 of the 'Magisteria' (Add MS 6782, f. 121).


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[Empty page]
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Rough working for page 15 of the 'Magisteria' (Add MS 6782, f.


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[Note:
Formulae for entries in the b, c, and d columns of a difference table, similar to those on page 14 of the 'Magisteria' (Add MS 6782, f. 121).


421211
[Note:
An incomplete version of the difference table on page 9 of the 'Magisteria' (Add MS 6782, f.


422211v
[Note:
Formulae for entries in the d, c, and b columns of a difference table; see page 16 of the 'Magisteria' (Add MS 6782, f. 123).


423212
[Note:
Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f.


Operatio. G
[tr: Working on G]
operatio. 1a
[tr: Working (1)]
operatio. 2a
[tr: Working (2)]
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Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f.


Residuum operationis. G
2a Working
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[Note:
General notation for triangular numbers.
See also page 2 of the 'Magisteria' (Add MS 6782, f.


3a notatio triangularium per notas
[tr: 3rd notation for triangular numbers, in general symbols.]
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[Note:
Square roots of binomes of the fifth and sixth kind by the general rule derived in Add MS 6788, f. 15 (and elsewhere). Here Harriot works with two types of fifth (bb+cc+b and bb+cd+b), according to whether the difference between the squares of the two terms is a square or not. Elsewhere he refers to these as bin. 5ʹ and bin. 5ʺ.
Similarly he distinguishes two types of sixth (bc+dd+bcb and bc+df+bc). Elsewhere he refers to these as bin. 6ʹ and bin. 6ʺ.
In all cases the roots are


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[Note:
Square roots of binomes of the third and fourth (bbc+bbc-ddc and b+bb-bd), by the general rule derived in Add MS 6788, f. 15 (and elsewhere). In both cases the roots are checked by


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[Note:
Square roots of binomes of the first and second (b+bb-cc and bb(bb-dd)+bb-dd), by the general rule derived in Add MS 6788, f. 15 (and elsewhere). In both cases the roots are checked by


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[Note:
Square roots of b+c, bbbb-bbdd+bbdd-dddd, and bbc+ddc, by the general rule derived in Add MS 6788, f. 15 (and elsewhere). In each case, the root is checked by multiplication. The numerical examples in Add MS 6783, f. 360v, f. 361, and Add MS 6782, f. 228, are closely related to the work on this page.


Nam: eius
[tr: For: its square]
Quia: duo
[tr: Because: two squares]
Et: duo
[tr: And: two rectangles]
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[Note:
Square roots of bb+cc+2bc, bbbb-dddd+2bbbd-2bddd, and bbc+ddc+4dc, by the general rule derived in Add MS 6788, f. 15 (and elsewhere). In each case, the root is checked by multiplication.


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Examinatio æquationis per
[tr: An examination of an equation in numbers]
et ita est (ut
[tr: and so it is (as above)]
et pro c
[tr: and for c]

et ita est (ut
[tr: and so it is (as below)]
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In
[tr: On Achilles]

vel per æquationem
[tr: or by the equality of ratios]

[tr: Another way]

Sit ratio motus ab, ad ac=co
[tr: Let the ratio of motion of ab to ac be co]

Tempus
[tr: Time; Time]

[tr: Another way]
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[Note:
The verses on this page describe the rules for operating with positive quantities ('more') and negative quantites ('lesse'). The first verse sets out the rules for multiplication. The second and third verses deal with subtraction of a negative quantity from a negative quantity, where the result may be either positive or negative.
Like folios Add MS 6784, f. 323, f. 324, which follow soon after it, this one appears to be based on In artem analyticen isagoge, 1591, in this case on Chapter IV, Praeceptum II and Praeceptum III.


If more by more must needes make more
Then lesse by more makes lesse of more
And lesse by lesse makes lesse of lesse
If more be more and lesse be
Yet lesse of lesse makes lesse or more
The which is best keep both in store
If lesse of lesse thou you wilt make lesse
Then pull bate the same from that is
But if the same thou you wilt make more
Then add the same to it to that is the sign of more
The signe rule of more is best to use
Except some Yet for some cause the do other choose then it refuse
For So Yet both are one, for both are true
of this inough and so
643322
[Note:
This page shows several examples of additions and subtractions using letters. Note that here such operations are only carried out between quantities of the same dimension.


1) Operationes logisticæ, in
[tr: The operations of arithmetic in symbols.]

[tr: add]


[tr: sum]

[tr: subtract]


[tr: remainder]
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645323
[Note:
This page shows examples of multiplication and division using



[tr: multiply]


[tr: by]


[tr: product]

[tr: divide]


[tr: by]


[tr: result]
manifestum
per præcog-
 nitam genera-

[tr: evident from the previously learned constructions]
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[Note:
The examples of division on this page are taken directly from In artem analyticen isagoge, 1591, Chapter IV, end of Praeceptum IV, but Harriot has re-written the examples in his own symbolic notation.


648324v
[Empty page]
649325
[Note:
The terminology and examples on this page are taken directly from In artem analyticen isagoge, 1591, Chapter V, but Harriot has re-written the examples in his own symbolic notation.



[tr: Let:]

Dico quod: per
[tr: I say that, by antihesis:]


[tr: Because:]

Adde
[tr: Add to each side.]


[tr: Therefore:]
Secundo:
[tr: Second, let:]

Dico
[tr: I say that:]


[tr: Because:]

Adde
[tr: Add to each side.]


[tr: Therefore.]

Et
[tr: And thus.]


[tr: Let.]

Dico quod. per
[tr: I say that, by hypobibasmus.]

[tr: Let.]

Dico quod: per
[tr: I say that, by parabolismus.]

Vel,
[tr: Or, let:]

dico
[tr: I say that.]
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[Note:
The reference to Apollonius is to pages 5 and 6 of Commandino's Apollonii Pergaei conicorum libri quattuor (1566). There are also references at the bottom of the page to Viète an
The reference to Viète is to Apollonius Gallus, Appendix 2, Problem
V. Dato triangulo, invenire punctum, a quo ad apices dati trianguli actæ tres lineæ rectæ imperatam teneant
Given a triangle, find a point from which there may be drawn three straight lines to the vertices of the given triangle, keeping a fixed ratio.
The reference to Cardano is to his Opus novum de proportionibus. The relevant Propositions are 154 (though mistakenly described in the Opus novum as 144) and 160.
Propositio centesimaquadragesimaquarta
Sint lineæ datæ alia linea adiungatur, ab extremitatibus autem prioris lineæ duæ rectæ in unum punctum concurrant proportionem habentes quam media inter totam & adiectam, ad adiectam erit punctus concursus a puncto extrema lineæ adiectæ distans per lineam mediam. Quod si ab extremo alicuius lineæ æqualis mediæ seu peripheria circuli cuius semidiameter sit media linea duæ lineæ ad prædicta puncta producantur, ipsæ erunt in proportione mediæ ad adiectam.
Hæc propositio est admirabilis:
Propositio centesimasexagesima
Proposita linea tribusque in ea signis punctum invenire, ex quo ductæ tres lineæ sint in proportionibus


5. Appolonius. pag. 5.
[tr: Apollonius, pages 5, 6.]
Quæsitum:
ubicunque signatur in periferia punctum h
erit; ah, hb : c, d: vel ak, kb
[tr: Sought:
Wherever a point h is placed on the circumference, then ah:hb=c:d or ak:kb]
sint data puncta a, b,
Data ratio. c. d.
producatur, ab, versus, f
[tr: Let the given points be a, b, the given ratio c:d. Let ab be produced towards f]
Dico
[tr: I say that:]
Inde: g maior, quam bf
minor, quam af
fiat fk=g
fiat kh periferia
sumatur quovis puncta h
Ducantur: ha, hb, hf
[tr: Whence, g is greater than bf, less than af. Let fk=g, let kh be the circumference, taking any point h. Let there be drawn ha, hb, hf]
* Ducantur bl, parallela, ah.
ubicunque signatur in periferia punctum h
erit; ah, hb : c, d: vel ak, kb
[tr: Taking bl parallel to ah, wherever the point h is placed on the circumference, then ah:hb=c:d or ak:kb]
Corollaria.
Hinc a tribus punctis sive sint in recta
vel non; possunt duci tres lineæ ad unum
punctum, ut s et erunt in data
[tr: Corollary
Hence from three points, whether in a straight line or not, it is possible to draw three lines to a single point, and they will be in the given ratio.]
vide vertam
in Apolonio gallo
et card: de prop. pag. 145.
[tr: see over, in Apollonius Gallus, and Cardano, De proportionibus, pages 145, ]
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[Note:
The reference is to pages 5 and 6 of Commandino's edition of Apollonii Pergaei conicorum libri quattuor


Ad appolonium. pa. 5.
[tr: On Apollonius, pages 5, 6]
Data puncta k, in linea, ab
Invenire lineam bf
ita ut sit:
Sit factum:

[tr: Given a point k in a line ab, find the line f so that:
Let it be done, ]
Aliter

[...]

sed idem ut
[tr: Another way

[...]

but the same as ]
Invenire fk
[tr: To find fk]
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[Note:
On this page Harriot investigates Proposition 18 from Supplementum geometriæ
Proposition XVIII.
Si duo triangula fuerint aequicrura singula, & ipsa alterum alteri cruribus aequalia, angulus autem qui est ad basin secundi sit triplus anguli qui est ad basin primi: triplum solidum sub quadrato cruris communis & dimidia base primi multata continuatave longitudine ejus cujus quadratum æquale est triplo quadrato altitudinis primi, cum multabitur ejusdem dimidiæ basis multatæ continuatve cubo, æquale est solido sub base secundi & ejusdem cruris
If two triangles are each isosceles, equal to one another in theri legs, and moreover the angle at the base of the second is three times the angle at the base of the first, then three times the product of the square of the common leg and half the base of the first decreased or increased by a length whose square is equal to three times the square of the altitude of the first, when reduced by the cube of the same half base thus decreased or increased, is equal to the product of the second base and the square of the common leg.
For Harriot's statement of Propostion 18, and a geometric version of the proof, see Add MS 6784, f. 349. Here he works the proposition algebraically.
This page also refers to Proposition 17 from the Supplementum, (see MS 6784, f. 350).


prop. 18.
[tr: Proposition 18 from the Supplementum]
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[Note:
On this page Harriot investigates Proposition 18 from Supplementum geometriæ
Proposition XVIII.
Si duo triangula fuerint aequicrura singula, & ipsa alterum alteri cruribus aequalia, angulus autem qui est ad basin secundi sit triplus anguli qui est ad basin primi: triplum solidum sub quadrato cruris communis & dimidia base primi multata continuatave longitudine ejus cujus quadratum æquale est triplo quadrato altitudinis primi, cum multabitur ejusdem dimidiæ basis multatæ continuatve cubo, æquale est solido sub base secundi & ejusdem cruris
If two triangles are each isosceles, equal to one another in theri legs, and moreover the angle at the base of the second is three times the angle at the base of the first, then three times the product of the square of the common leg and half the base of the first decreased or increased by a length whose square is equal to three times the square of the altitude of the first, when reduced by the cube of the same half base thus decreased or increased, is equal to the product of the second base and the square of the common leg.
This page refers to several previous propositions from the Supplementum, namely Proposition 12 and 14b (Add MS 6784, f. 353), Proposition 16 (add MS 6784, f. 351) and Proposition 17 (add MS 6784, f.


prop. 18.
[tr: Proposition 18 from the Supplementum]
Si duo triangula fuerint aequicrura singula, et ipsa alterum alteri cruribus aequalia; angulus
autem qui est ad basin secundi sit triplus anguli qui est ad basin primi. Triplum solidum
sub quadrato cruris communis, et dimidia base primi multata continuatave longitudine
ejus cujus quadratum æquale est triplo quadrato altitudinis primi, cum multabitur ejusdem
dimidiæ basis multatæ continuatve cubo, æquale est solido sub base secundi et ejusdem
cruris
[tr: If two triangles are each isosceles, equal to one another in their legs, and moreover the angle at the base of the second is three times the angle at the base of the first, then three times the product of the square of the common leg and half the base of the first decreased or increased by a length whose square is equal to three times the square of the altitude of the first, when reduced by the cube of the same half base thus decreased or increased, is equal to the product of the second base and the square of the common ]
Sit triangulum primum ABC, secundum
CDE. quorum crura et anguli sint
ut exigit propositio. Et sit GB dupla
BF. Tum quadratum GF erit triplum quadrati
BF  

[tr: Let the first triangle be ABC and the second CDE, whose sides and angles are as specified in the proposition. And let GB be twice BF. Then the square of GF is three times the square of BF]
Nam:
per 15,p
[...]
Hoc est, in notis proportionalium quas notum 12,p
1o. Ducantur omnia per AG

[...]

Hoc est in notis
[tr: For by Proposition 15
[...]
that is, in the notation for proportionals noted in Proposition 12,
1. Multiply everything by AG.

[...]

That is, in the notation of Proposition ]
2o. Ducantur omnia per CG

[...]

Hoc est in notis
[tr: 2. Multiply everything by CG.

[...]

That is, in the notation of Proposition ]
Deinde per 16.p
Hoc est in notis 12,p.
Sed: per consect: 14.p
Ergo patet
[tr: Thence by Proposition 16,
That is, in the notation of Proposition 12
But by the consequence of Proposition 14,
Thus the propostion is ]
Cum 16a et 17a prop. basis AC notabatur (a) ideo eius partes
Scilicet AG et GC alijs vocalibus notandæ sunt. pro AG nota (e)
et pro GC, (o). AB et CE servent easdem notas quas ibi
habuerunt. Videlicet AB, (b) et CE, (c).
Propositum igitur simplicibus notis ita
[tr: Since in Propositions 16 adn 17, the base AC is denoted by a, therefore its parts, namely AG and GC may be denoted by other names; for AG put the letter e and for GC the letter o. For AB and CE use the same notation as they had there, namely AB=b and CE=c.
In simple notation the proposition may therefore be ]
igitur:
Quando æquatio est sub ista
forma:
a erit duplex vel. AG. vel. GC
[tr: When the equation is in this form, a is twofold, either AG or GC]
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[Note:
On this page Harriot investigates Proposition 17 from Supplementum geometriæ
Proposition XVII.
Si duo triangula fuerint aequicrura singula, & ipsa alterumalteria cruribus aequalia, angulus autem, quem is qui est ad basin secundi relinquit e duobus rectis, sit triplus anguli qui est ad basin primi: solidum triplum sub base primi & cruris communis quadrato, minus cubo e base primi, aequale est solido sub base secundi & cruris communis
If two triangles are each isosceles, both with equal legs, and moreover the angle at the base of the second subtracted from two right angles is three times the angle at the base of the first, then three times the product of the base of the first and the square of the common side, minus the cube of the first base, is equal to the product of the second base and the square of the common
The working contains reference to three propositions from Euclid's Elements
II.6 If a straight line be bisected and produced to any point, the rectangle contained by the whole line so increased, and the part produced, together with the square of half the line, is equal to the square of the line made up of the half, and the produced part.
III.36 If from a point without a circle two straight lines be drawn to it, one of which is a tangent to the circle, and the other cuts it; the rectangle under the whole cutting line and the external segment is equal to the square of the
I. 47 In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the


prop. 17.
[tr: Proposition 17 from the Supplementum]
Si duo triangula fuerint aequicrura singula,
et ipsa alterumalteria cruribus aequalia; angulus
autem, quem is qui est ad basin secundi relinquit
e duobus rectis, sit triplus anguli qui est ad basin
secundi primi. Solidum triplum sub base primi et cruris
communis quadrato, minus cubo e base primi: aequale
est solido sub base secundiet cruris communis

[tr: If two triangles are each isosceles, the legs of one equal to the legs of the other, and moreover the angle at the base of the second is three times the angle at the base of the first, then the cube of the first base, minus three times the product of the base of the first and the square of the common side, is equal to the product of the second base and the square of the same ]
per 6,2 el.
per 36,3 el.
per 47,1 el.

[...]

quia parallogramma æquialta
et sunt ut bases.

[...]

vel per notas
simplices
Hæque Resoluatur Analogia, erit:

[tr: by Elements II.6
by Elements III.35
by Elements I.47

[...]

because the parallelograms are of equal height and are as the bases.

[...]

or in simple notation
And this ratio is resolved, hence the ]
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[Note:
On this page Harriot investigates Proposition 16 from Supplementum geometriæ
Proposition XVI.
Si duo triangula fuerint aequicrura singula, & ipsa alterum alteri cruribus aequalia, angulus autem qui est ad basin secundi sit triplus anguli qui est ad basin primi: cubus ex base primi, minus triplo solido sub base primi & cruris communis quadrato, aequalis est solido sub base secundi & ejusdem cruris
If two triangles are each isosceles, the legs of one equal to the legs of the other, and moreover the angle at the base of the second is three times the angle at the base of the first, then the cube of the first base, minus three times the product of the base of the first and the square of the common side, is equal to the product of the second base and the square of the same side.
The working contains a reference to Euclid's Elements, Proposition
II.5 If a straight line be divided into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half that line.


prop. 16.
[tr: Proposition 16 from the Supplementum]
Si duo triangula fuerint aequicrura singula,
et ipsa alterum alteri cruribus aequalia: angulus
autem qui est ad basin secundi sit triplus
anguli qui est ad basin primi. Cubus ex
base primi, minus triplo solido sub base primi
et cruris communis quadrato, aequalis
est solido sub base secundi et ejusdem
cruris
[tr: If two triangles are each isosceles, the legs of one equal to the legs of the other, and moreover the angle at the base of the second is three times the angle at the base of the first, then the cube of the first base, minus three times the product of the base of the first and the square of the common side, is equal to the product of the second base and the square of the same ]
per 5,2 el.

[...]

Quia parallogramma æquialta
et sunt ut bases. BH. HD.

[...]

vel per notas
simplices
Resoluatur analogia et erit:

[tr: by Elements II.5

[...]

Because the parallelograms are of equal height and are as the bases BH, HD.

[...]

or in simple notation
The ratio is resolved, and hence the ]
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On this page Harriot investigates Proposition 15 from Supplementum geometriæ
Proposition XV.
Si e circumferential circuli cadant in diametrum perpendiculares duæ, una in centro, altera extra & ad perpendicularem in centro agatur ex puncto incidentiæ perpendicularis alterius, linea recta faciens cum diametro angulum æqualem trienti recti; a puncto autem quo acta illa secat perpendiculare in centro, ducatur alia linea recta ad angulum semicirculi: triplum quadratum huius, æquale est tam quadrato perpendicularis quae incidit extra centrum, quam quadratis segmentorum diametri, inter quæ perpendicularis illa media est
If from the circumference of a circle there fall two perpendiculars onto the diameter, one to the centre, the other off-centre; and to the perpendicular to the centre there is drawn from the point of incidence of the other perpendicular a straight line making an angle equal to one-third of a right angle to the diameter; moreover from the point where that line cuts the perpendicular to the centre, there is drawn another line to the angle of the semicircle, then three times the square of it is equal to the square of the perpendicular which falls off-centre and the squares of the segments of the diameter between which the perpendicular is the mean
The working contains a reference to Euclid's Elements, Proposition
II.4 If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the parts, together with twice the rectangle contained by the


prop. 15.
[tr: Proposition 15 from the Supplementum]
Si e circumferential circuli cadant in
diametrum perpendiculares duæ; una in
centro; altera extra centrum: et ad per-
 pendicularem in centro agatur ex puncto
incidentiæ perpendicularis alterius, linea
recta faciens cum diametro angulum æqualem
trienti recti, a puncto autem quo acta illa secat
perpendiculare in centro, ducatur alia
linea recta ad angulum semicirculi; Triplum
quadratum huius, æquale est tam quadrato perpendicularis quae incidit extra centrum,
quam quadratis segmentorum diametri, inter quæ perpendicularis illa media est

[tr: If from the circumference of a circle there fall two perpendiculars onto the diameter, one to the centre, the other off-centre; and to the perpendicular to the centre there is drawn from the point of incidence of the other perpendicular a straight line making an angle equal to one-third of a right angle to the diameter; moreover from the point where that line cuts the perpendicular to the centre, there is drawn another line to the angle of the semicircle, then three times the square of it is equal to the square of the perpendicular which falls off-centre and the squares of the segments of the diameter between which the perpendicular is the mean proportional.]

Sit diameter circuli ABC, a cuius circumferentia cadat perpendiculariter DB et fit
AB minus segmentum, BC maius, E verum centro. Sed et cadat quoque e circumferentia
perpendiculariter FE, et ex B ducatur recta BG ita ut angulus GBE sit æqualis trienti
recti, unde fiat BG dupla ipsius GE; et iungatur AG. Dico triplum quadratum ex AG
æquari quadrato ex DB, una cum quadrato ex AB et quadrato ex BC
[tr: Let ABC be the diameter of a circle, from whose circumference there falls perpendicularly DB, and let AB be the lesser segment, BC the greater, and E the centre. But there also falls perpendicularly from the circumference FE, and from B there is drawn a line BG so that the angle GBE is equal to a third of a right angle, whence BG is twice GE; and AG is joined. I say that three times the square on AG is equal to the square on DB together with the square on AB and the squareon BC]

Etiam
per 4,2 El.

[...]
Addatur utrovisque

[...]
Ergo

[tr: Also by Elements II.4

[...]

Hence the ]
Hinc tale Consectarium potest
[tr: Here a Consequence of this kind may be inferred]
Datis tribus continue proportionalibus: invenire lineam cuius
quadratum sit tertia pars adgregati quadratorum e tribus

[tr: Given three continued proportionals, find a line whose square is a third of the sum of the squares of all three ]
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[Note:
On this page Harriot investigates Propositions 12, 13, and 14 from Supplementum geometriæ
Proposition XII.
Si fuerint tres lineæ rectæ proportionales: cubus compositæ e duabus extremis, minus solido quod fit sub eadem composita & adgregato quadratorum a tribus, æqualis est solido sub eadem composita & quadrato
If there are three proportional lines, the cube of the sum of the two extremes, minus the product of that sum and the sum of squares of all three, is equal to the product of the sum and the square of the
Proposition XIII.
Si fuerint tres lineæ rectæ proportionales: solidum sub prima & adgregato quadratorum a tribus, minus cubo e prima, æquale est solido sub eadem prima & adgregato quadratorum secundæ &
If there are three proportional lines, the product of the first and the sum of squares of all three, minus the cube of the first, is equal to the product of the first and the sum of squares of the second and third.
Proposition XIV.
Si fuerint tres lineæ rectæ proportionales: solidum sub prima & adgregatum quadratorum a tribus, minus cubo e tertia, æquale est solido sub eadem tertia & adgregato quadratorum primæ &
If there are three proportional lines, the product of the first and the sum of squares of all three, minus the cube of the third, is equal to the product of the third and the sum of the first and second.
The 'Consectarium' appears verbally in Viete's proposition; Harriot has re-written it in symbolic


prop. 12.
[tr: Proposition 12 from the Supplementum]
Si fuerint tres lineæ rectæ proportionales: cubus compositæ e duabus extremis,
minus solido quod fit sub eadem composita et adgregato quadratorum a tribus:
æqualis est solido sub eadem composita et quadrato
[tr: If there are three proportional lines, the cube of the sum of the two extremes, minus the product of that sum and the sum of squares of all three, is equal to the product of the sum and the square of the second.]

Sint 3 continue proportionales
utrinque addatur

[...]

Fiant solida ab extremis et etiam a medijs, et inde:

[tr: let there be three continued proportionals
add to each side

[...]

There may be made solids from the extremes and also form the means, and hence the ]
Prop. 13. Si fuerint tres lineæ rectæ proportionales: solidum sub prima et adgregato
quadratorum tribus, minus cubo e prima: æquale est solido sub eadem
prima et adgregato quadratorum secundæ et
[tr: Proposition 13. If there are three proportional lines, the product of the first and the sum of squares of all three, minus the cube of the first, is equal to the product of the first and the sum of squares of the second and ]

Sint tres continue proportionales

[...]

Resoluatur Analogia et erit:

[tr: Let there be three continued proportionals

[...]

The ratio is resolved, and hence the ]
Prop. 14. Si fuerint tres lineæ rectæ proportionales: solidum sub prima et adgregatum quadratorum
a tribus minus cubo e tertia: æquale est solido sub eadem tertia et adgregato
quadratorum primæ et
[tr: Proposition 14. If there are three proportional lines, the product of the first and the sum of squares of all three, minus the cube of the third, is equal to the product of the third and the sum of the first and ]

Sint tres continue proportionales

[...]

Resoluatur Analogia et erit:

[tr: Let there be three continued proportionals

[...]

The ratio is resolved, and hence the ]

[tr: Consequence]
Quia æquantur æqualibus
ex antecedente
[tr: Because equals are equated to equals, by the preceding conclusion.]
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[Note:
On this page Harriot investigates Propositions 10 and 11 from Supplementum geometriæ
Proposition X.
Si fuerint tres lineæ rectæ proportionales: est ut prima ad tertiam, ita adgregatum quadratorum primæ & secundæ ad adgregatum quadratorum secundæ &
If there are three proportional lines, as the first is to the third, so is the sum of squares of the first and second to the sum of squares of the second and third.
Proposition XI.
Si fuerint tres lineæ rectæ proportionales: est ut prima ad adgregatum primae & tertiæ, ita quadratum secundæ ad adgregatum quadratorum secundæ &
If there are three proportional lines, as the first is to the sum of the first and third, so is the square of the second to the sum of squares of the second and third.
There are two references to Euclid's Elements, Proposition
VI.20 Similar polygons my be divided into the same number of similar triangles, each similar pair of which are proportional to the polygons; and the polygons are to each other in the duplicate ratio of their homologous
The 'Consectarium' appears verbally in Viete's proposition; Harriot has reinterpreted it


prop. 10.
[tr: Proposition 10 from the Supplementum]
Si fuerint tres lineæ rectæ proportionales: Est ut prima ad tertiam, ita adgregatum
quadratorum primæ et secundæ ad adgregatum quadratorum secundæ et
[tr: If there are three proportional lines, as the first is to the third, so is the sum of squares of the first and second to the sum of squares of the second and ]

sint tres proportionales
continue
consequetur
vel
Et per synæresin
Et per 20,6 Euclid
Ergo pro
[tr: let there be three continued proportionals
consequently
or
And by synæresis
And by Euclid VI.20
Therefore in ]
prop.
[tr: Proposition 11]
Si fuerint tres lineæ rectæ proportionales, est ut prima ad adgregatum primae et
tertiæ, ita quadratum secundæ ad adgregatum quadratorum secundæ et
[tr: If there are three proportional lines, as the first is to the sum of the first and third, so is the square of the second to the sum of squares of the second and ]

sint tres proportionales
per 20,6 El
Et per Synæresin

[tr: let there be three proportionals
by Elements VI.20
And by synæresin
It may be ]

[tr: Consequence]
Itaque si fuerint tres lineæ rectæ proportionales, tria solida ab ijs
effecta æqualia sunt. per 10am conculsionem
per 11am conclu.

[...]

Dua prima solida sunt æqualia, quia unum factum est ab extremis analogia 10am
et alterum a modijs. Tertium est factum a modijs inferioris analogia 11am,
cuius extremæ sunt eædem superioris analogia 10am, et illo æquale.
[tr: Therefore if there are three lines in proportion, three solids constructed from them are equal.
by the conclusion of the 10th
by the conclusion of the 11th

[...]

The two first solids are equal, because one is made from the extremes of the ratio of the 10th, and the other by the method
The third is made by the method of the ratio of the 11th, whose extremes are the same as in the ratio of the 10th, and is equal to that one.]
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On this page Harriot examines a particular case arising from Proposition VII of Supplementum geometriæ (1593), when the fourth proportional is twice the first. The same proposition is the subject of Chapter V of Viète'sVariorum responsorum libri VIII, which was also published in
Caput V
Propositio
Describere quatuor lineas rectas continue proportionales, quarum extremæ sint in ratione
Construct four lines in continued proportion, whose extremes are in double
The text in the Variorum refers to the Supplementum, indicating that the Supplementum was written


Ad Corollorium prop. 7. Supplementi. Et ad cap. 5. Resp. lib. 8. pag.
[tr: On a corollary to Proposition 7 of the Supplement. Also Chapter 5, Variorum liber responsorum, page ]
Sit AB prima proportionalium, et BC ea
cuius quadratum est triplum quadrati AB.
Tum AC est dupla ad AB; et per assumptum
ex poristicis in alia charta demonstratum AC
erit quarta proportionalis. Per propositione EA est secunda et EG tertia.
Sed FB est æqualis EG propter similitudine triangulorum EFB et EAC, et
analogiam precedentam ut sequitur.
AB.EA.EG.AC. Analogia precedens.

[...]

Et per similitudi-
 num Δorum
[...]

Ergo. AB.AE.FB.AC. continue
[tr: Let AB be the first proportional, and BC that whose square is three times the square of AB.
Then AC is twice AB; and by taking it from the proof demonstrated in the other AC will be the fourth proportional. By the proposition EA is the second and EG the third.
But FB is equal to EG because of similar triangles EFB and EAC, and
the precding ratio, as follows.
AB:EA:EG:AC preceding ratio.

[...]

And by similar triangles.

[...]

Therefore AB:AE:FB:AC are continued ]
[Note: The other sheet mentioned in this paragraph appears to be Add MS 6784, f. 356. ]
Datis igitur extremis in ratione dupla, mediæ ita compendiosæ

[tr: Therefore given the extremes in double ratio, the mean is briefly found.]

Sit maxima AC bisariam divisa in puncto D et intervallo DC describatur
circulus. Et sit prima minima AB inscripta et producta ad partes E.
Ducatur CE ita ut EF sit æqualis AB. et acta fit linea FB.
Quatuor igitur continue proportionales ex supra demonstratis
[tr: Let the maximum AC be cut in half at the point D and with radius DC there is described a circle. And let the minimum AB be inscribed and produced to the point E. Construct CE so that EF is equal to AB, and let the line FB be joined.
Therefore there are the four continued proportionals that were demonstrated ]
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[Note:
On this page Harriot examines a particular case arising from Proposition VII of Supplementum geometriæ (1593), when the fourth proportional is twice the


prop. 7. Supplementi de
[tr: Proposition 7 of the Supplement, on a corollary]
Sint 4or proportionales
in specie.
Si quarta sit dupla ad prima, erit:

[...]

Ergo quatuor proportionales
quarum extremæ sunt in
ratione dupla
[tr: Let there be 4 proportionals in general form.
If the fourth is twice the firs, then:

[...]

Therefore the four proportionals whose extremes are in double ratio will ]
Tunc fac
[...]
et nota quadratorum
[tr: Then make [the square of the first and second and the square of the third and fourth], and note the difference of the squares.]
Differentia quadratorum 3,bb
Hoc est triplum quadratum primæ
[tr: The difference of the squares is 3bb.
This is three times the square of the first ]
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[Note:
This page investigates the proposition that is the subject of Chapter V of Variorum responsorum libri VIII. It appears to be a continuation of Add MS 6784, f. 355.
Caput V
Propositio
Describere quatuor lineas rectas continue proportionales, quarum extremæ sint in ratione
Construct four lines in continued proportion, whose extremes are in double


In Cap. 5. Resp. lib. 8. pag.
[tr: Chapter 5, Variorum liber responsorum, page 4.]
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[Note:
On this page Harriot examines Proposition VII from Supplementum geometriæ
Propositio VII.
Data è tribus propositis lineis rectis proportionalibus & ea cujus quadratum æquale fit ei quo differt quadratum compositae ex secunda & tertia à quadrato compositæ ex secunda & prima, invenire secundam & tertiam
Given the first of three proposed proportional straight lines, and another whose square is equal to the difference between the square of the sum of the second and third, and the square of the sum of the second and first, find the second and third


prop. 7.
[tr: Proposition 7 of the Supplement]
Data e tribus propositis lineis rectis proportionalibus prima et ea
cujus quadratum aequale fit ei quo differt quadratum compositae ex
secunda et tertia a quadrato compositæ ex secunda et prima: invenire
secundam et tertiam
[tr: Given the first of three proposed proportional straight lines, and another whose square is equal to the difference between the square of the sum of the second and third, and the square of the sum of the second and first, find the second and third proportionals.]
Data prima AB
Et recta BC
[tr: The first given line AB and the straight line BC]
Tum tres proportionales

[tr: Then the three proportionals will be:]
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a)
Sit a, Achilles.
t,
[tr: Let a be Achilles, t the ]
Sit ratio motus a, ad motus t,
ut: b, ad c.
nempe: 10 ad
[tr: Let the ratio of the motion of a to the motion of t be as b to c, namely, 1 to ]
Et sit distantia a, et t. d. nempe 1 mille
[tr: And let the distance between a and t be d, namely, one thousand ]
Et sit motus utriusque in eadem linea et ad easdem partes, nempe
ab a, et t versus w
[tr: And suppose the motion of both is in the same line and in the same direction, namely, from a and t towards w]
Quæritur ex datis punctum ubi Achilles comprehendet
[tr: From what is given there is sought the point where Achilles catches up with the tortoise.]
Quæestio solvitur exhibendo summam infinitæ progressionis decrescentis
ut sequitur: (species summa infinitæ progressionis decrescentis
ut in doctrinam de proggeom:
[tr: The problem is solved by producing the sum of an infinite decreasing progression as follows: (the case of the sum of an infinite decreasing progression as in the teaching of geometric porgressions ]
Alia
[tr: Other progressions.]
(ut Archimedes de
quad: parab: pr:
[tr: (as Archimedes in the quadrature of the parabola, proposition 23)]
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b)
Sit (a), Achilles.
t,
[tr: Let a be Achilles, t the ]
Sit velocitas motus a, ad velocitatem motus t,
ut: b, ad c
[tr: Let the speed of motion of a to the speed of motion of t be as b to c]
Sit distantia inter (a) et (t). d
[tr: Let the distance between a and t be d]
Et sit motus utriusque in eadem linea et ad easdem partes, nempe ab (a), et (t)
versus w
[tr: And let the mtion of both be in the same line and the same direction, namely from a and t towards w]
Quæritur ex datis punctum ubi Achilles comprehendet
[tr: From what is given there is sought the point where Achilles catches up with the tortoise.]
Ponatur illud punctum esse w. et sit tw. e
[tr: Suppose this point is w, and let the distance tw be e]

Datur igitur e. Et inde w
[tr: Therefore e is found; and hence w]
In numeris sit b. 10.
c. 2.
d. 2.
[tr: In numbers let b=10, c=2, d=2 ]

[tr: Another way.]
Aliter 2o.
Quæritur aw et sit y
[tr: A second way.
There is sought aw, and suppose it is y]
Exemplum de duabus [¿]numeribus[?]
[tr: An example from two numbers.]
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[Note:
On this folio, Harriot derives the sum of a finite geometric progression, using Euclid V.12 and its numerical counterpoart, Euclid VII.12. He then extends his result to an infinite (decreasing) progression, by arguing that the final term must be infnitely small, that is, nothing.
Euclid V.12: If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.
Euclid VII.12: If there be as many numbers as we please in proportion, then, as one of the antecedents is to one of the consequents, so are all the antecedents to all the


1.) De progressione
[tr: On geometric porgressions]

[tr: Theorem]

el. 5. pr:
[tr: Elements, Book 5, Proposition ]

el. 7. pr.
[tr: Elements, Book 7, Proposition ]

Si sint magnitudines quotcunque proportionales, Quemadmodum
se habuerit una antecedentium ad unam consequentium: Ita
se habebunt omnes antecedentes ad omnes
[tr: If any number of magnitudes are proportional, then just as as one antecedent is to its consequent, so will the sum of the antecedents be to the sum of the consequents.]
Sint continue proportionales. b, c, d, f, g, h
[tr: Let the continued proportionals be b, c, d, f, g, h]
In notis universalibus
[tr: In general notation we have]

p. primum. p. primus terminus
[tr: p. first term. p. first term of the ]

s. secunda. s.
[tr: s. second. s. ]

u.
[tr: u. ]

o.
[tr: o. ]
Ergo; si, p > s ut in progressi
[tr: Therfore if p > s are in a decreasing ]
Ergo; si, p < s ut in progressi
[tr: Therfore if p > s are in an increasing ]
De infinitis progressionibus
decrescentibus in
[tr: For a progression descreasing indefinitely:]

Cum progressio decrescit et
numerus terminorum sit infinitus;
ultimus terminus est infinite
minimus hoc est nullius
[tr: Since the progression decreases and the number of terms is infinite, the last term is infnitely small, that is, of no ]


[tr: Therefore.]
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[Note:
In the preceding folio, f. 361, Harriot derived a formula for the sum of a finite geometric progression based on Euclid V.12. Here he gives an alternative derivation based on Euclid IX. 35.
Euclid IX. 35: If as many numbers as we please be in continued proportion, and there be subtracted from the second and the last numbers equal to the first, then as the excess of the second is to the first, so will the excess of the last be to all those before it.


2.) De progressione
[tr: On geometric porgressions]

[tr: Theorem]

el. 9. pr:
[tr: Elements Book IX, Proposition ]

Si sint quotlibet numeri deinceps proportionales, detrahuntur autem
de secundo et ultimo æquales ipsi primo: erit quemadmodum
secundi excessus ad primum, ita ultima excessus ad omnes qui ultimum

[tr: If there are as many numbers as we please in proportion, and the first is subtracted from the second and the last, then just as the difference of the second is to the first, so is the difference of the last to all before the ]
Progressio
[tr: An increasing progression:]
In notis universalibus: sit p, primus: s, secundus: u, ultimus: o,
[tr: In general notation, let p be the first term; s the second term; u the last term; o the ]
Progressio
[tr: A decreasing progression:]
In notis universalis
[tr: In general notation we have:]
Vel: in notis magis universalis.
sit p, primus terminus rationis. s, secundus.
M, maxumus terminus progressionis
m, minimus.
[tr: Or, in more general notation, let p be the first term of the ratio, s the M the greatest term of the progression, m the least. ]
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[Note:
In this folio Harriot repeats statements that are to be found in Variorum responsorum, Chapter XVII (1646, 397–398).
Harriot's letters M, m, o, M, m correspond to Viete's D, X, F, D, B.
Harriot's final comments refer to the final sentence of Viete's penultimate paragraph (1646, 398):
Et ut differentia terminorum rationis ad terminorum rationis majorem, ita maxima ad compositam ex ombnibus plus cremento.

[tr: As the difference in the terms of the ratio is to the greater term of the ratio, so is the the greatest term of the progression to the sum plus an ]


3.) De progressione geometrica. (ut Vieta in var:
[tr: On geometric progressions (as Viete in Variorum responsorum]

[tr: Increasing.]

[tr: Decreasing.]
m. minor terminus
[tr: Let m be the lesser terms of the ]

M. Maior terminus
[tr: Let M be the greater terms of the ]

M. maximus terminus
[tr: Let M be the greatest term of the ]

m. minimus terminus
[tr: Let M be the least term of the ]

o. omnes, id est summa
[tr: o is all, that is the sum of ]
ita Vieta post δεδόμενα
in respons: pag.
[tr: thus Viete after δεδόμενα inVariorum Responsorum page ]
apud Vieta dicitur
[tr: in Viete this is said to be the increment.]
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[Note:
On this folio an expression that looks like p=s is to be read as |p-s|


De progressionibus.
finitis &
[tr: On finite and infinite progressions]
linea infinite longaquælibet = æqualis alicui, plano
solido.
longo-solido.
plano-solido.
solido-solido. &
[tr: An infinite line of any length is equal to some plane, or solid, or solid-length, or solid-plane, or solid-solid, etc.]
linea infinite brevis quælibet = æqualis alicui, puncto.
linea.
puncto-plano.
puncto-solido. &c.

[tr: Any infinitely short line is equal to some line-point, or plane-point, or solid-point, etc.]

Quælibet punctum terminat
[tr: Whatever point terminates the progression.]
infinite numero puncta = lineæ
plano.
solido. &
[tr: an infinite number of points equal a line, or plane, or solid, etc.]

linea signata
terminat
progressionem.
ita planum
[tr: a designated line terminates the progression; similarly a designated plane,]
hæc & alia huius generis

[tr: these and others of this kind may be considered.]
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[Note:
This page contains a symbolic version of Euclid Book II, Proposition 11:
II.11. To cut a given straight line so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment.


propositiones 2i
[tr: Propositions from the second book of Euclid]
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1.) De reductione
[tr: On the reduction of equations]
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1)B) De reductione
[tr: On the reduction of equations]
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[Note:
Here Harriot solves the equation 25=6a-aa (in modern notation, 25=6x-x2) for the roots 3+-1 and 3--1. He then checks by multiplication that these valus do indeed satisfy the equation.


803402
[Note:
Powers of (20+4) up to (20+4)5 following the pattern laid out in Add MS 6782, f. 276.
A calculation below each box gives the sum of the figures contained in


804402v
[Note:
The calculations from the previous page (Add MS 6784, f. 402) are checked by root


The extraction
of the
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[Note:
Third, fourth, and fifth powers of (20 + 4).
The binomial coefficients 3, 3 and 4, 6, 4 and 5, 10, 10, 5, appear amongst the numbers in the rightmost column.


808404v
The doctrine of Algebraycall nombers is but
the doctrined of such continuall proportionalles of
which a unite is the
A unite being the first of continuall proportionalles; the second is
called a roote: because the third wilbe always a square: & the fourth
third a cube, as Euclide The names of the other proportionalles
following are all compounded of squares, or cubes or both according
to Diophantus & others which follow Some or other of the most parte of the later
writers gave the name of surdsolidus, of which the first or simple sursolid
is the sixt proportionall. &
Any nomber may be any terme proportinall in a continuall progression
from a If the nomber terme be the second, the third is gotten by
multiplying the nomber into him & the fourth by multiplying the
third by the second & so as also by the doctrine of progression
any terme that is found another may be gotten compendiously
without continuall
If a nomber that is known & designed to be the third, fourth,
or fifth or any other proportinall of another denomination: the
doctrine to find the second is that which is called the extraction
of the roote, which is taught in these
The second proportionall is also called the first dignity, & the third the
second dignity, & the fourth the third dignity &
The third is also called the first power; the 4th the second power &
The first proportionall
is a
The first dignity is
the second proportionall,
called a
The first power is the
third proportionall
called a square
or second Dignity
called a
The first solid is the
fourth proprtionall:
The third dignity: &
The second power,
called a
The pythagoreans
did call 4 the first solid
as Boethius
The nomber serveth to be, because pyramides are prime solids
& 4 amongst nombers is the first
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[Note:
Here Harriot demonstrates that multiplication by 9 increases the number of digits by one as far as the 21st power but not at the 22nd power. Thus the number of digits alone is no guide to the size of the


An induction to prove that
to pricke the second figure for
the extraction of square rootes
& the third for cubes & 4th
for biquadrates etc. according
to the nomber of figures that
the greatest figure 9 doth
produce is no for we
may see how it breaketh in
the 22th proportionall dignity & so
but the true case
of such pricking appeareth
out of the speciosa genesis which
is in an other paper
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[Note:
Calculation of powers of c+b+a to show how the digits of a three-digit number are distributed in the


If the roote to be extracted be three figures
the two first as one may here see are to be had
according to the generall rule, the next is
also to be gotten really after the same manner
that is supposing the two first to be as one, & that
which foloweth, the second; although in appearance
& expressing by wordes it seems
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[Note:
Powers of (b–c) and (bb+bc+cc)


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[Note:
Here and on folio Add MS 6784, f. 412, Harriot shows that the product of two or three unequal parts is always less than the product of the same number of equal parts.


1o. de
[tr: 1. on bisection]
Sit: tota linea. 2b.
vel duæ æquales partes. b+b.
magnitudo facta ab illis
erit quadratum bb
[tr: Let the total line be 2b
or two equal parts b+b,
the size of their product will be the square bb]

Sint inæquales partes. b+c
et: b-c
[tr: Let there be unequal parts b+c and b-c]

magnitudo facta: bb-cc<bb
[tr: the size of the product is bb-cc<bb]
Si linea dividatur utcunque in tot
partes inæquales, quot æquales:
Magnitudo facta ab inæquali-
 bus, minor est illa quæ facta
ab
[tr: If a line is divided in any way into as many unequal parts as equal parts, the size of the product of the unequal parts is less than the product of the equal ]

[tr: or:]

Si aggregatum linearum inæqualium æqueretur
aggregato tot æqualium: Magnitudo facta &
[tr: If the sum of the unnequal lines is equal to the sum of as many equals, the size of the product etc.]

[tr: also:]

plana facta ab inæqualibus
minora sunt quaduratis
facta ab
[tr: planes made from unequals are less than squares made from equals.]
2o. De sectione in tres
[tr: 2. On sectioning into three parts.]
Casus
[tr: First case.]

Sint tres inæquales
[tr: Let there be three unequalparts.]

magnitudo facta: bbb-bcc
[tr: the size of the product is bbb-bcc]
Tres æquales
[tr: Three equal parts.]

magnitudo facta
quæ cubus. bbb>bbb-bcc
[tr: the size of the product which is a cube is bbb>bbb-bcc]
Casus 2a
[tr: Case 2.]

Sint tres inæquales
[tr: Let there be three unequal parts.]

magnitudo facta. bbb+3,bbc
[tr: the size of the product is bbb+3bbc]
Tres æquales
[tr: Three equal parts.]

magnitudo facta
quæ cubus. bbb+3,bbc+3,bcc+ccc>bbb+3,bbc
[tr: the size of the product which is a cube is bbb+3bbc+3bcc+ccc>bbb+3bbc]
822411v
[Note:
Note the combinations of g (greater than), l (less than), and e (equals), and of the symbols <, >, = in the lower part of the


823412
[Note:
The continuation of Add MS 6784, f.


Casus 3a
[tr: Case 3.]

Sint tres inæquales
[tr: Let there be three unequal parts.]

magnitudo facta. bbb-3,bbc
[tr: the size of the product is bbb-3bbc]
Tres æquales
[tr: Three equal parts.]

magnitudo facta
quæ cubus. bbb-3,bbc+3,bcc-ccc>bbb+3,bbc
[tr: the size of the product which is a cube is bbb-3bbc+3bcc-ccc>bbb-3bbc]
Casus 4a
[tr: Case 4.]

Sint tres inæquales
[tr: Let there be three unequal parts.]

magnitudo facta. bbb+3,bbd-9,bcc-9,bdc
[tr: the size of the product is bbb+3bbd-9bcc-9bdc]
Tres æquales
[tr: Three equal parts.]

magnitudo facta
quæ cubus. bbb+3,bbd+3,bdd+ddd>bbb+3,bbd-9,bcc-9,bdc
[tr: the size of the product which is a cube is bbb+3bbd+3bdd+dd>bbb+3bbd-9bcc-9bdc]
Casus 5a.
et
[tr: Case 5, and last.]

Sint tres inæquales
[tr: Let there be three unequal parts.]

magnitudo facta. bbb-3,bbd-9,bcc+9,bcd
[tr: the size of the product is bbb-3bbd-9bcc+9bcd]
Tres æquales
[tr: Three equal parts.]

magnitudo facta
quæ cubus. bbb-3,bbd+3,bdd-ddd>bbb-3,bbd-9,bcc+9,bcd
[tr: the size of the product which is a cube is bbb-3bbd+3bdd-ddd>bbb-3bbd-9bcc+9bcd]
nam: 3,bdd+9,bcc>ddd+9bcd.
[tr: for: 3bdd+9bcc>ddd+9bcd]
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[Note:
Combinations of small numbers; see also Add MS 6784, f.


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[Note:
This page summarizes in shorthand some rules that are written out in full in Harriot's treatise on cubic equations, on Add MS 6782, f. 186.
The abbreviations 'co:l' and 'co:pl' stand for 'longitudinal coefficient' and 'plane coefficient' respectively. In an equation of the form aaa-baa+cca=ddf, the longitudinal coefficient is b and the plane coefficient is cc. Below the diagram Harriot has set out the different conditions under which such an equation can have three real roots, not necessarily distinct. The same sets of roots are also listed in Add MS 6783, f. 281.
The relevant equations are worked in full in sheets marked C, D, E, F, G (Add MS 6782, f. 315, f. 315v, f. 317, f. 318, f. 319), and also in Add MS 6783, f. 185.


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Ad generationes sequentium specierum
[tr: On the generation of the following types of equation.]
Æquatio substantiva

[tr: Parabolic equation]
Æquatio adiectiva hyperbolica
sive additiva. Hyperbolic equation
Æquatio ablativa elliptica
sive
[tr: Elliptic, or Bombelli's, equation]
Ergo æquatio nullitatis prima
sive [???]
sive
[tr: Therefore the equation is primitive.]

Ergo verum quod
[tr: Therefore what was proposed is true.]
Ad resolutiones sequentium specierum
[tr: On solving the following types of equation]

æquatio
[tr: parabolic equation]

æquatio
[tr: hyperbolic equation]

æquatio
[tr: elliptic equation]




prinatus.
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The polynomial aaa-3abb evaluated for a=b, a=2b, a=3b, ... , a=7b. The resulting coefficients of bbb are listed in the table at the bottom of the page. Columns to the right list successive differences as far as the constant difference 6. The table has also been extrapolated upwards, giving rise to negative values in the first three columns. There is an error in the first column, however, which reading upwards should be: 322, 110, 52, 18, 2, - 2, 0, 2,


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Note various combinations of small numbers in the lower part of the page (see also Add MS 6784, f.


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[Note:
Sums of some infinite geometric


856428v
[Note:
Triangles and circles filled with rectilinear figures (rectangles or triangles), in a way that can in principle be continued indefinitely.


857429
De infinitis. Ex ratione motus, temporis et
[tr: On infinity. From the ratio of motion, time and space.]
Vide Arist. lib. 6. tret. 23.
proclum de motu lib. 1. pro.
[tr: See Aristotle, Book 6, Treatise 23.
Proclus, De motu, Book 1, Proposition ]
1.
Moveatur A corpus
per bc spatium in
tempore de atque sit
ille motus
[tr: Let a body A be moved through a distance bc in a time de and let that motion is ]
infinite

[tr: infinite maximum]

[tr: minimum]

[tr: an indivisible]

[tr: a point]
aliquod
infinite

[tr: infinite maximum]
minimum
eadem

[tr: minimum in the same ratio]
Indivisibile
eadem

[tr: An indivisble in the same ratio]
Indivisibile
sed non punctum
vel instans ut alia
ratione
[tr: And indivisble but not a point or an instant that can be inferred from the other ratio.]
2.
Moveatur A corpus per bc spatium
in tempore de2 atque sit ille
motus
[tr: Let a body A be moved thorugh a distance bc in time de2 and let that motion be ]

[tr: an indivisible]

[tr: a point]
Indivisibile
eadem
[tr: An indivisble in the same ratio]
Indivisibile quod
dimidium est
Indivisibilis ex
priori
[tr: An indivisble whose half is indivisble by the previous argument.]
Ergo
[tr: Therefore also]
Indivisibile quod
dimidium est
Indivisibilis ex
priori
[tr: An indivisble whose half is indivisble by the previous argument.]

[tr: a point]

[tr: a point]
Ergo punctum quod ponebatur esse
indivisbile, alia ratione inferetur
Divisibile, et sic in
[tr: Therefore a point that can be supposed indivisble, is inferred from the other ratio to be divisible, and thus ]
858429v
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859430
[Note:
Triangles transformed to spirals.
See also Add MS 6785, f. 437 and Add MS 6784, f. 246, f. 247, f.


860430v
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861431
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862431v
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