כוןכחרוןאפ
ARTES
181.7
SCIENTIA
VERITAS
LIBRARY
OF THE
UNIVERSITY OF MICHIGAN
ΣR PLURIEUS
TUE BOR
TY
SI-QUÆRIS PENINSULAM-AMŒNAME
CIRCUMSPICE
THILLENNI
NON
CANTAL, PRAFEL NEC NON ACADEUSDEM PROTANG
I.Mynde fc.
Revelidee
رہا کر
EUCLIDE's ELEMENTS;
The whole
FIFTEEN BOOKS,
compendiouſly Demonſtrated:
WITH
ARCHIMEDES's Theorems of the
Sphere and Cylinder, Inveſtigated by the
Method of Indivifibles.
ALSO,
EUCLIDE's DATA,
and a brief
Treatife of REGULAR SOLIDS.
By ISAAC BARROW, D.D. late Master of
Trinity College in Cambridge.
The whole carefully Corrected, and Illuftrated
with Copper Plates.
To which is now added an
APPENDIX,
Containing,
The Nature, Conftruction, and Application of
Logarithms.
By J. BARROW, Author of Navigatio Britannica, &c.
LONDON: Printed for W. and J. MOUNT, and T. PAGE
on Tower-Hill; and C. HITCH and L. HAWES in Pater-n‹fter-
Row; R. MANBY and S. Cox on Ludgate-Hill; E. COMYNS
under the Royal-Exchange; J. and J. RIVINGTON in St. Paul's
Church Yard; and J. WARD in Cornhill, oppofite the Royal-
Exchange, 1751.
Histje. To the READER.
Brand
***
F you are defirous, Courteous Reader, to know
what I have performed in this Edition of the
ELEMENTS Of EUCLIDE, I shall here explain it to
you in fhort, according to the Nature of the Work.
I have endeavoured to attain two Ends chiefly; the
first, to be very perfpicuous, and at the fame time fo
very brief, that the Book may not fwell to fuch a Bulk,
as may be troublefome to carry about one, in both which
I think I have fucceeded. Some of a brighter Genius,
and endued with greater Skill, might have demonftrated
moſt of theſe Propofitions with more nicety, but per-
haps none with more fuccinctness than I have; efpeci-
ally fince I alter'd nothing in the Number and Order
of the Author's Propofitions; nor prefum'd either to.
take the Liberty of rejecting, as less neceffary, any of
them, or of reducing fome of the easier fort into the
· Rank of Axioms, as feveral have done; and among
others, that most expert Geometrician A. Tacquetus C.
(whom I the more willingly name, becauſe I think it is
! but civil to acknowledge that I bave imitated him in
fome Points) after whofe most accurate Edition I had
no Thoughts of attempting any thing of this Nature,
'till I confidered that this most learned Man thought fit
to publish only Eight of EUCLIDE's Books, which he
took the pains to explain and embellish, having in a
manner rejected and undervalued the other Seven, as
lefs appertaining to the Elements of Geometry. But
my Province was originally quite different, not that
of writing the Elements of Geometry after what method
foever I pleas'd, but of demonftrating, in as few Words
as poſſible I could, the whole Works of EUCLIDE. As
аг
to
*
To the READER.
to Four of the Books, viz. the Seventh, Eighth, Ninth,
and Tenth, although they do not fo nearly appertain to
the Elements of plain and folid Geometry, as the fixe
precedent and the two fubfequent, yet none of the more
Skilful Geometricians can be fo ignorant as not to know
that they are very useful for Geometrical Matters, not
only by reason of the very near affinity that is be-
tween Arithmetick and Geometry, but alſo for the Know-
ledge of both commenfurable and incommenfurable Mag-
nitudes, fo exceeding necessary for the Doctrine of both
plain and folid Figures. Now the noble Contempla-
tion of the five regular Bodies that is contained in the
three laft Books, cannot, without great Injustice, be pre-
termitted, fince that for the fake thereof our soixeins,
τοιχειωτής,
being a Philofopher of the Platonic Sect, is faid to bave
compos'd this univerſal System of Elements; as Proclus lib.
2. witneſſeth in thefe Words, "Olev dny tãs ovjerάons
τοιχειώσεως τέλα προεςήσατο τὴν τῶν καλεμένων
πλαγωνικών χημάτων σύςασιν. Befides, I eafily per fuaded
y felf to think, that it would not be unacceptable to any
Lover of thefe Sciences to have in his Poffeffion the whole
Euclidean Work, as it is commonly cited and celebra-
ted by all Men: Wherefore I refolved to omit no Book
or Propofition of thofe that are found in P. Herigo-
nius's Edition, whofe Steps I was obliged clofely to
follow, by reafon I took a Reſolution to make use of most
of the Schemes of the faid Book, very well forefeeing that
Time would not allow me to form new ones, though
fometimes I chofe rather to do it. For the fame Reafon
I was willing to use for the most part EUCLIDE's own
Demonftrations, having only exprefs'd them in a more
fuccinit Form, unleſs perhaps in the Second, Thirteenth,
and very few in the Seventh, Eighth, and Ninth
Books, in which it ſeem'd not worth my while to de-
viate in any Particular from him: Therefore I am not
without
To the READER.
without great hopes, that as to this Part I have in
fome meaſure fatisfied both my own Intentions, and the
Defire of the Studious. As for fome certain Problems
and Theorems that are added in the Scholions (or short
Expofitions) either appertaining (by reason of their
frequent Ufe) to the Nature of thefe Elements, or con-
ducing to the ready Demonftration of thofe Things that
follow, or which imitate the Reasons of fome prin-
cipal Rules of Practical Geometry, reducing them to
their original Fountains, thefe I fay, will not, I hope,
make the Book fwell to a Size beyond the defign'd
Proportion.
The other Butt which I levell'd at, is to content
the Defires of those who are delighted more with fym-
bolical than verbal Demonftrations. In which Kind,
whereas most among us are accuftom'd to the Symbols
of Gulielmus Oughtredus, I therefore thought beft to
make ufe, for the most part, of his. None hitherto
(as I know of) has attempted to interpret and publiſh
EUCLIDE after this manner, except P. Herigonius;
whofe Method (tho' indeed most excellent in many
things, and very well accommodated for the particular
purpoſe of that most ingenious Man) yet it ſeems in my
Opinion to labour under a double Defect. First, in
regard, that, altho' of two or more Propofitions produ-
ced for the Proof of any one Problem or Theorem, the
former do not always depend on the latter, yet it do not
readily enough appear, either from the order of each,
or by any other manner, when they agree together,
and when not; wherefore, for want of the Conjunctions
and Adjectives, ergo, rurfus, &c. many difficulties
and occafions of doubt often arise in reading, eſpecially
to thoſe that are Novices. Befides it frequently hap-
pens, that the faid Method cannot avoid fuperfluous
Repititions, by which the Demonftrations are often-
times
To the READER.
times render'd tedious, and fometimes also more intri-
cate; which Faults my Method easily removes by the
arbritary mixture of both Words and Signs: There-
fore let what has been faid, touching the Intention
and Method of this little Work, fuffice. As to the
reft, whoever covets to please himſelf with what may
be ſaid, either in Praise of the Mathematicks in gene-
ral, or of Geometry in particular, or touching the Hi-
Story of thefe Sciences, and confequently of EUCLIDE
bimfelf, (who digefted thofe Elements) and others
3alremoì of that kind, may confult other Interpreters.
Neither will I (as if I were afraid left thefe my En-
deavours may fall fhort of being fatisfactory to all Per-
fons) alledge as an Excufe (though I may very lawfully
do it) the want of due time which ought to be employ'd
in this Work, nor the Interruption occafioned by other
Affairs, nor yet the want of requifite help for theſe
Studies, nor feveral other things of the like Nature.
But what I have bere employ'd my Labour and Study
in for the Ufe of the ingenious Reader, I wholly
Submit to his Cenfure and Judgment, to approve if
useful, or reject if otherwiſe,
4
1
I, B.
Thus
To the READER.
Thus far the Learned Author. But as the work
bas been often printed fince his death, and by that means
feveral errors committed, I have, at the request of the
Bookfellers concerned in this treatiſe, and from a fincere
respect and veneration for the memory of the deceafed
Author, carefully reviſed the whole performance; and
flatter myself, from the great pains and care I have taken,
that very few errors will be found in this Edition.
And as the wooden cuts in the former editions, were,
by often printing, almoft obliterated, their place is now
fupplied by figures engraven on Copper Plates, and pro-
per care taken to correct the inaccuracies and errors
committed by cutting them on Wood; which has given
this edition great advantages over the former, both with
regard to beauty and correctness.
In the Appendix which I have added to this work,
I have endeavoured to render the construction and uſe of
Logarithms as plain and eafy as poffible. And that no-
thing might be wanting for underſtanding the nature of
thoſe tables used in trigonometrical calculations, I have
added an inveſtigation of the feveral feries invented by the
illuftrious Sir Ifaac Newton, for finding the length of
the circumference of the circle, in equal parts of the
radius, alfo of the fine, tangent and fecant of any
arch, in the fame parts; with the application of thefe
feries to the conftructing the triangular canon, and the
quadrature of the Circle. I have also fhewn the manner
of computing the artificial or logarithmic fines, tangents,
and fecants, from the length of the arch of the circle
first given in equal parts of the radius, independant of
the tables of logarithms, fines, tangents, and fecants.
J. BARROW.
•
+
X
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༢
The Explication of the Signs or
Signifies
Equal.
Greater.
Leffer.
Characters.
More, or to be added.
Lefs, or to be ſubtracted.
The Differences, or Excefs; Alſo, that all
the Quantities which follow, are to be
fubtracted, the Signs not being changed.
Multiplication, or the Drawing one fide of
a Rectangle into another.
The fame is denoted by the Conjunction
of Letters; as AB=AxB.
Continued Proportion.
The Side or Root of a Square, or Cube, &c.
✓
Q & q
A Square.
C & C
A Cube.
The Ratio of a fquare Number to a ſquare
Number.
Other Abbreviations of Words, where-ever they
occur, the Reader will without trouble understand
of himself; ſaving ſome few, which, being of lefs
general ufe, we refer to be explained in their Places,
most commonly at the beginning of each Book in
which they are ufed.
1
The
DE
[
The FIRST BOOK
O F
EUCLIDE'S
ELEMENT S.
f.
A
Definitions.
Point is that which hath no parts.
II. A Line is a longitude without
latitude.
III. The ends, or limits of a line
are points.
IV. A Right-line is that which lies
equally betwixt its points.
V. A Superficies is that which hath
only longitude and latitude.
VI. The extremes, or limits of a fuperficies are lines.
VII. A Plain-fuperficies is that which lies equally be-
twixt its lines.
VIII. A Plain-angle is the inclination of two lines
the one to the other, the one touching the other in the
fame plain, yet not lying in the fame ftrait-line.
IX. And if the lines which contain the angle, be right.
lines, it is called a right-lined angle.
A
X. Whea
2
The first Book of
Plate I.
Fig. 1.
Fig. 2.
X. When a right-line CG, ftanding upon a right-line
AB, makes the angles on either fide thereof, CGA, CGB,
equal one to the other, then both thoſe equal angles are
right-angles; and the right-line CG, which ftandeth on
the other, is termed a Perpendicular to that (AB) where-
on it ftandeth.
Note, When Several angles meet at the fame point (as at
G) each particular angle is deſcribed by three letters; wherè-
of the middle letter fheweth the angular point, and the two
other letters the lines that make that angle: As the angle
which the right-lines CG, AG make at G, is called CGA,
or AGC.
XI. An Obtufe-angle is that which is greater than a
right-angle; as DGB.
XII. An Acute-angle is that which is lefs than a right-
angle; as DGA.
XIII. A Limit, or Term, is the end of any thing.
XIV. A Figure is that which is contained under öne
or more terms.
XV. A Circle is a plain figure contained under one
line, which is called a circumference; unto which all
lines, drawn from one point within the figure, and falling
upon the circumference thereof, are equal the one to
the other.
XVI. And that point is called the center of the circle.
XVII. A Diameter of a circle is a right-line drawn
thro' the center thereof, and ending at the circumference
on either fide, dividing the circle into two equal parts.
XVIII. A Semicircle is a figure which is contained
under the diameter and that part of the circumference
which is cut off by the diameter.
In the circle EABCD, E is the center, AC the diameter,
ABC the femicircle.
XIX. Right-lined figures are fuch as are contained
under right-lines.
XX. Three-fided or trilateral figures are fuch as are
contained under three right-lines.
XXI. Four-fided or quadrilateral figures are fuch as
are contained under four right-lines.
XXII. Many-fided figures are fuch as are contained
under more right-lines than four.
XXIII. Of trilateral figures, that is, an equilateral
triangle, which hath three equal fides; as the triangle
ABC. Fig. 3.
XXIV.
EUCLIDE's Elements.
3
XXIV. An Ifofceles, is a triangle which hath only two
fides equal; as the triangle ABC. Plate I. Fig. 4.
XXV. A Scalenum, is a triangle whoſe three fides are
all unequal; as ABC. Fig. 5.
XXVI. Of theſe trilateral figures, a right-angled tri-
angle is that which hath one right-angle, as the triangle
ABD. Fig. 5.
XXVII. An Amblygonium, or obtufe-angle triangle,
that is which hath one angle obtufe; as ABC. Fig. 5.
XXVIII An Oxgonium, or acute-angled triangle, is
that which hath three acute angles; as ABC. Fig. 4.
An equiangular, or equal-angled figure is that whereof
all the angles are equal. Two figures are equiangular,
if the feveral angles of the one figure be equal to the
feveral angles of the other. The fame is to be under-
ftood of equilateral figures.
XXIX. Of Quadrilateral, or four-fided figures, a
fquare is that whofe fides are equal, and angles right;
as ABCD Fig. 6.
XXX. A Figure on the one part longer, or a long
fquare, is that which hath right-angles, but not equal
fides; as ABCD. Fig. 7.
XXXI. A Rhombus, or diamond-figure, is that which
has four equal fides, but is not right-angled; as ABCD.
Fig. 8.
XXXII. A Rhomboides, is that whofe oppofite fides,
and oppofite angles are equal; but has neither equal nor
right angles; as ABCD. Fig. 9.
XXXIII. All other quadrilateral figures befides theſe
are called trapezia, or tables; as GNDH. Fig. 10.
XXXIV. Parallel, or equi-diftant right-lines are fuch,
which being in the fame fuperficies, if infinitely produced,
would never meet as AD and BC. Fig. 11.
A 2
XXXV,
4
The first Book of
XXXV. A Parallelogram is a quadrilateral figure, whofe
oppofite fides are parallel, or equi-diftant; as ABCD.
Plate I. Fig. 7.
XXXVI. In a Parallelogram AGEL, Fig. 9. when a
diameter AE, and two lines BK, CF, parallel to the
fides, cutting the diameter in one and the fame point D,
are drawn, ſo that the Parallelogram be divided by them
into four Parallelograms; thofe two LD, DG, through
which the diameter does not pafs, are called comple-
ments; and the other two CB, KF, through which the
diameter paffeth, the Parallelograms ftanding about the
diameter.
A Problem is, when fomething is propoſed to be done or
effected.
A Theorem is, when ſomething is propoſed to be demon-
Strated.
A Corollary is a Confectary, or ſome confequent truth
gained from a preceding demonftration.
A Lemma is the demonftration of fome premife, whereby
the proof of the thing in hand becomes the shorter.
1.
FR
Poftulates or Petitions.
Rom any given point to any other given point,
to draw a right-line.
To produce a finite right-line, ftraït forward conti-
nually.
3. Upon any center, and at any diſtance, to deſcribe
a circle.
Axioms.
I
Τ TH
Hings equal to the fame thing, are alfo equal
one to the other.
As A B C. Therefore AC; or therefore all
A,B,C, are equal the one to the other.
Note, When feveral quantities are joyned the one to the
other continually with this mark, the firft quantity is by
virtue of this axiom equal to the last, and every one to every
one: In which cafe we often abstain from citing the axiom,
for brevity's fake; altho' the force of the confequence depends
thereon.
2. If to equal things you add equal things, the wholes
will be equal.
3. If from equal things you take away equal things,
the things remaining will be equal.
4. If
1
EUCLIDE's Elements.
5
in
4. If to unequal things you add equal things, the
wholes will be unequal.
5. If from unequal things you take away equal things,
the remainders will be unequal.
6. Things which are double to the fame third, or to
equal things, are equal one to the other. Underſtand
the fame of triple, quadruple, &c.
7. Things which are half of one and the fame thing,
or of things equal, are equal the one to the other. Con-
ceive the fame of fubtriple, fubquadruple, &c.
8. Things which agree together, are equal one to the
other.
The converfe of this axiom is true in right-lines and an-
gles, but not in figures, unless they be like.
Moreover, magnitudes are faid to agree, when the parts of
the one being apply'd to the parts of the other, they fill up an
equal or the fame place.
9. Every whole is greater than its part.
10. Two right-lines cannot have one and the fame
fegment (or part) common to them both.
11. Two right-lines meeting in the fame point, if they
be both produced, they fhall neceffarily cut one another
in that point.
12. All right-angles are equal the one to the other.
13. If a right-line EF (Plate 1. Fig. 11.) falling on
two right-lines, AD, BC, make the internal angles on
the fame fide, GFE, FEG, lefs than two right-angles,
thoſe two right-lines produced fhall meet on that fide
where the angles are lefs than two right-angles.
14. Two right-lines do not contain a ſpace.
15. If to equal things you add things unequal, the
exceſs of the wholes fhall be equal to the excess of
the additions.
16. If to unequal things equal be added, the excefs
of the wholes fhall be equal to the excefs of thoſe
which were at firſt.
17. If from equal things unequal things be taken
away, the excess of the remainders fhall be equal to the
exceſs of what was taken away.
18. If from things unequal things equal be taken
away, the excess of the remainders fhall be equal to the
excess of the wholes.
19. Every whole is equal to all its parts taken to-
gether.
A 3
201
{
6
The first Bock of
a 3. poſt.
b 1. poff.
c 15. def.
15.def.
C
d 1. ax.
€ 23, def.
20. If one whole be double to another, and that which
is taken away from the firft be double to that which is
taken away from the fecond, the remainder of the firſt
ſhall be double to the remainder of the fecond.
The Citations are to be understood in this manner; When
you meet with two numbers, the firft fhews the Propofition,
the fecond the Book.; as by 4. 1. you are to underſtand the
fourth Propofition of the firft Book; and fo of the reſt,
Moreover, ax. denotes Axiom, poft. Poftulate, def. Definition,
fch. Scholium, cor. Corollary.
U
PROPOSITION I. Plate I. Fig. 12.
Pon a finite right-line given as AB, to defcribe an
equilateral triangle AČB.
From the centers A and B, at the diftance of AB,
or BA, (a) defcribe two circles cutting each other in
the point C; from whence (b) draw two right-lines
CA, CB. Then is AC (c) AB (c) = EC (d)=AC,
(e) Wherefore the triangle ACB is equilateral. Which
was to be done.
a 3. post.
b 1. poft.
C I. I.
d 2. poſt.
e 2. post.
f 15. def.
g constr.
h 3 ax.
k 15. def.
1. 1. ax.
Scholium.
After the fame manner upon the line AB may be de
ſcribed an Iſofceles triangle, if the diſtances of the
equal circles be taken greater or leſs than the line AB,
PRO P. II. Fig. 13.
From a point given A, to draw a right-line AG equal to a
right-line given BC.
From the center, at the distance of CB, (a) defcribe
the circle CBE. (b) Join AC; upon which (c) raiſe the
equilateral triangle ADC. (d) Produce DC to E. From
the center D, at the diſtance of DE, deſcribe the circle
DEG; and let DA (e) be produced to the point G in the
circumference thereof. Then AGCB.
For DG (f)DE, and DA (g) = DC.
DC. Wherefore
AG (h) =CE (k) — BC (1). Which was to be done.
The putting of the point A within or without the line
BC varies the caſes; but the conſtruction, and the de-
monſtration, are every where alike,
Schol.
EUCLIDE's Elements.
7
Schol.
The line AG might be taken with a pair of com-
paffes; but the fo doing anfwers to no poftulate, as
Proclus well obferves.
PRO P. III. Plate I. Fig. 14.
Two right-lines, A and DC, being given, from the
greater DC, to take away the right-line DE, equal to
the lefer A.
From the point D (a) draw the right-line B D A.
The circle defcribed from the center D at the diſtance
of BD fhall cut off DE (b) BD (c) = A (d)
DE.
Which was to be done.
PRO P. IV. Fig. 15, and 16.
=
If two triangels BAC, EDF, have two fides of the one
BA, AC, equal to two fides of the other ED, DF, each to
its correfpondent fide (that is BA-ED, and AC=DF) and
have the angel A equal to the angle D contained under the
equal right-lines; they shall have the bafe BC equal to the
bafe EF; and the triangle BAC shall be equal to the triangle
EDF; and the remaining angels B, C, fhall be equal to the
remaining angles, E, F, each to each, under which the equal
Jides are fubtended.
If the point D be apply'd to the point A, and the
right-line DE plac'd upon the right-line AB, the point
E fhall fall upon B, becauſe DE (a)=AB, alfo the right-
line DF fhall fall upon AC, becauſe the angle A (a)—D.
Moreover the point F ſhall fall upon the point C, becauſe
AC (a) DF. Therefore the right-lines EF, BC, ſhall
agree, becauſe they have the fame terms, and confequently
are equal. Wherefore the triangles, BAC, DEF, and the
angles B, E, as alfo the angles C, F, do agree, and are
equal. Which was to be demonftrated.
PROP. V. Fig. 4
The angles BAC, BCA, at the base of an Ifofceles tri-
angle ABC, are equal one to the other, And if the equal
fides AB, BC, are produced, the angles CAD, ACE, under
the bafe, fhall be equal one to the other.
A 4
(a) Tak
a 2. I.
b 15. df.
c conftr.
d 1. ax.
a hype
8
The first Book of
a 3. 1.
b 1.post.
c hyp.
d conftr.
f
e 4. I.
3. ax
g 4.1.
h before.
k 3. ax.
23. I.
b 1. poſt.
c Suppos.
d byp
€ 4. I.
f
£ 9. ax.
a 9 ax.
Fig. 17.
b 5. 1.
c fuppas.
(a) Take BE BD; and (b) join CD, and AE.
Becaufe, in the triangles BCD, BAE, are AB (c) =
BC, and BE, (d) BD, and the angle B common to
them both, (e) therefore is the angle BAE BCD,
and the angle AEB (e) =BDC, and the baſe AE (c)=
CD; alfo EC (ƒ) = DA. Therefore in the triangles AEC,
ADC (g) will be the angle ECA DAC, Which was to
be dem. Alfo the angle EAC DCA, but the angle
BAE (b) BCD; therefore the angle BAC (k) —
BCA. Which was to be demonftrated.
Coroll.
ས་
Hence, every equilateral triangle is alſo equiangular.
PRO P. VI. Plate I Fig. 4.
If two angles BCA, ACB of a triangle ABC, be equal
the one to the other, the fides BC, AB, fubtcnded under the
equal angles, ſhall alſo be equal one to the other.
If the fides be not equal, let one be bigger than the
other, ſuppoſe BABC. (a) Make AF BC, and (6)
draw the line CF.
In the triangles FAC, BAC, becauſe AF (c)=CB, and
the fide, AC is common, and the angle BAC (d)
ACB,
the triangles FAC, ACB (e) ſhall be equal the one to the
other, a part to the whole, (f) Which is impoſſible.
Caroll.
Hence, every equiangular triangle is alfo equilateral,
PROP. VII. Fig. 4.
Upon the fame right-line AC two right-lines being drawn
AB, BC, trvo other right-lines equal to the former, AF, CF,
each to each (viz. AF AB, and CF-BC) cannot be drawn
from the fame points A, C, on the fame fide B, to ſeveral
points, as B and F, but only to B.
1. Cafe. If the point F be fet in the line AB, it is
plain that AF is (a) not equal to AB.
2. Cafe. If the point D be placed within the triangle
ACB, then draw the line CD, and produce BDF, and ECF.
Now you would have AD—AC, then the angle ALC (1)
ACD; as alfo, becauſe BD (c) BC, the angle FDC=
(6)
EUCLIDE's Elements.
9
) ECD, therefore is the angle FDC— (4) ACD, that d9 ax.
is, the angle FDC—ADC. ‍(d) Which is impoffible.
3. Cafe. If D (Fig. 18.) falls without the triangle
ACB, let CD be joined.
Again, the angle ACD (e) —ADC, and the angle BCD
(c)=BDC. (f) Therefore the angle ACDBDC, viz.
the angle ADCBDC. Which is impoffible. Therefore,
&c.
PROP. VIII. Plate I. Fig. 15, and 16.
If two triangles ABC, DEF have two fides AB, AC,
equal to trvo fides DE, DF, each to each, and the baſe BC
equal to the baje EF, then the angles contained under the equal
right-lines fhall be equal, viz. À to D.
e 5. 1.
f 9. ax.
a hyp.
bax. 8.
c hyp.
Becaufe, BC (a) =EF, if the baſe BC be laid on the
baſe EF, (b) they will agree: therefore whereas AB (c)—
DE, and AC=DF, the point A will fall on D (for it can-
not fall on any other point, by the precedent propofition)
and fo the fides of the angles A and D are coincident;
(d) wherefore thoſe angles are equal. Which was to be de- d 8. ax.
monftrated.
Coroll.
1. Hence, triangles mutually equilateral are alfo mu-
tually (x) equiangular.
2. Triangles mutually equilateral (x) are equal one to
the other.
PROP. IX. Fig. 19.
To biject, or divide into two equal parts, a right-lined
angle given BAC.
(a) Take AD to AE, and draw the line DE; upon
which (6) make an equilateral triangle DFE, draw the
right-line AF; it ſhall biſect the Angle.
For AD (c) AE, and the fide AF is common, and the
bafe DF (c) FE. (d) therefore the angle DAFEAF.
Which was to be done,
Coroll.
Hence it appears how an angle may be cut into 4,
8, 16, 32, &c. equal parts, to wit, by bifecting each
part again.
The method of cutting angles into any number of
equal parts required, by a Rule and Compafs, is as yet
unknown to Geometricians,
PROP,
x 4. 1.
a 3. 1.
bi. I.
c conftr.
d 8. 1.
10
The first Book of
a 1. I.
9. 1.
b
c confir.
d 4. 1.
a 3. 1.
b 1. I.
c conftr.
d 8. 1.
e 10. def.
a 3 poft.
b 10. I.
c conftr.
d 8. I.
e 10. def.
PROP. X. Plate I. Fig. 20.
To bifect a right-line given AB.
Upon the line given AB (a) erect an equilateral tri-
angle ABC; and (b) bifect the angle C with the right-
line CD. That line ſhall alſo biſect the line given AB.
For AC (c) BC, and the fide CD is common, and
the angle ACD (c) BCD. therefore AD (d)=BD.
Which was to be done.
The practice of this and the preceding propofition is
eafily fhewn by the Conſtruction of the 1ft propofition of
this Book.
PROP. XI. Fig. 20.
From a point Ð in a right-line given AB to erect a right-
line DC at right-angles.
(a) Take on either fide of the point given DA=DB,
upon the right-line AB (6) erect an equilateral triangle,
draw the line CD, and it will be the perpendicular re-
quired.
For the triangles DAC, DCB are mutually (c) equila-
teral, (d) therefore the angle ADC=CDB ; (e) therefore
DC is perpendicular. Which was to be done.
The practice of this and the following is eafily per-
formed by the help of a fquare.
PROP. XII. Fig. 20.
Upon an infinite right-line given AB, from a point given
that is not in it, to let fall a perpendicular right-line CD.
From the center C (a) defcribe a circle cutting the
right-line given in the points A and B. Then (b) bifect
AB in D, and draw the right-line in CD, which will be
the perpendicular required.
Let the lines CA, CB be drawn. The triangles ADC
CDB are mutually (c) equilateral; (d) therefore the an-
gles ADC, BDC are equal, and by (e) confequence
right. (e) Wherefore DC is a perpendicular. Which was
to be done.
PROP. XIII. Fig. 1.
When a right-line DG ftanding upon a right-line AB
maketh angles DGA, DGB; it maketh either two right-
angles, or two angles equal to two right-angles.
?
EUCLIDE'S Elements.
11
If the angles DGA, DGB be equal, (a) then they
make two right-angles; if unequal, then from the point
G (b) let there be erected a perpendicular GC. Becauſe
the angle DGB (c) to a right-angle-+-CGD, and the
angle DGA (d) to a right-angle-DGC; therefore the
angle DGB--DGA (e) to two right-angles. Which
was to be demonftrated.
་
=
Corollaries.
1. Hence, if one angle DGB be right, the other DGA
is alſo right; if one acute, the other is obtufe, and ſo on
the contrary,
2. If more right-lines than one ftand upon the fame
right-line at the fame point, the angles fhall be equal to
two right.
3. Two right-lines cutting each other make angles
equal to four right-angles.
4. All the angles made about one point make four
right-angles; as appears by Coroll. 2.
PROP. XIV. Plate I. Fig. 1.
If to any right-line CG, and a point therein G two right-
lines, not drawn from the fame fide, do make the angles
CGA, CGB, on each fide equal to two right, the lines,
AG, GB, ſhall make one firait-line.
If you deny it, let AG, GF make one right-line; then
fhall the angle CGA+CGF (a) =two right angles (6)=
CGA+CGB. Which is (c) abfurd.
PRO P. XV. Fig. 21.
If tavo right-lines AB, CD, cut thro' one another, then are
the two angles which are oppofite, viz. CEB, AED, equal one
to the other.
For the angle AEC CEB (a)
angles
+
to two right right-
AECAED; (b) therefore CEB AĔD.
Which was to be demonftrated.
Schol. 1.
If to any right-line AB, and in it a point E, two right
lines being drawn CE, ED, and not taken on the fame
fide, make the vertical (or oppofite) angles CEA, and
BED equal, thoſe right-lines CE, ED, do meet directly
and make one ſtrait line.
For
a def. 10.
bii. I.
c 19. ax.
d 3. ax.
e 2. ax.
a 13. 1.
b hyp.
cg.ax.
a 13. 1.
b 3. ax.
12
The first Book of
2 13. I.
b 2. ax.
C 14..I
For two right-angles are (a) equal to the angle CEA+
CEB, (6)=CEBBED. (c) Therefore CE, ED, are
in a trait line. Which was to be demonftrated.
Schol. 2. Plate I. Fig. 21.
If four right-lines EA, EB, EC, ED, proceeding from
one point E, make the angles, vertically oppofite,
equal the one to the other, each two lines, AE, EB,
and CE, ED, are placed in one ſtrait line.
For becauſe the angle AEC AED + CEB + DEB
a 4. c 13. I. (a) to four right-angles, therefore the angle AEC
b kyp. &
AED (6) CEB+DEB to two right-angles. (c)
Therefore CED and AEB are ftrait-lines. Which was
to be demonftrated.
C2. ax.
C 14. I.
a 10. 1. &
1. poft.
b 3. 1.
c conftr.
₫ 15. 1
€ 4. I.
f15. I.
$9.ax.
2 13. 1.
b 16. 1.
£ 4. ax.
=
PRO P. XVI. Fig. 22.
One fide, BC, of any triangle ABC being produced, the
outward angle ACD will be greater than either of the
inward and oppofite angles, CAB, CBA.
Let the right-lines AH, BE, (a) bife&t the fides AC,
BC; from which lines produc'd, take (6) EFBE,
and HI, (b) =AH, and join FC, and IC; and pro-
duce ACG.
Becauſe CE (c) EA, and EF (c) EB, and the
angle FEC (BEA, the angle ECF (e) fhall be equal
to EAB. By the like argument is the angle ICH
ABH. Therefore the whole angle ACD (f) (BCG)
(g) is greater than either the angle CAB or ABC. Which
was to be demonftrated.
PRO P. XVII.
Fig. 23.
Two angles of any triangle ABC, which way foever they
are taken, are less than two right-angles.
Let the fide BC be produced. Becauſe the angle
ACD + ACB (a) — two right-angles, and the angle
ACD (b) ˜˜A, (c) therefore A+ ACB than two
right-angles. After the fame manner is the angle B +
ACB than two right-angles. Laftly, the fide AB
being produced, the angle AB will be alfo lefs than
two right-angles. Which was to be demonftrated.
Coroll
EUCLIDE's Elements,
13
Coroll.
1. Hence it follows that in every triangle wherein
one angle is either right or obtuſe, the two others are
acute angles.
2. If a right-line AE make unequal angles with ano-
ther right-line BC, one acute AEB, the other obtuſe
AEC, a perpendicular AB, let fall from any point A to
the other line BC, ſhall fall on that fide the acute angle
is of.
For if AC, drawn on the fide of the obtufe-angle, be
a perpendicular, then in the triangle AEC, fhall the
angles AEC ACE be greater than two right-angles.
* Which is contrary to the precedent prop.
3. All the angles of an equilateral triangle, and the
two angles of an Ifofceles triangle that are upon the
bafe, are acute.
PRO P. XVIII. Plate I. Fig. 23.
The greatest fide AC of every triangle ABC fubtends the
greatest angle ABC.
From AC (a) take away AF AB, and join BF.
(b) Therefore is the angle AFB ABF. But
AFB (c) CC; therefore is ABF C; (d) there-
fore the whole angle ABC C. After the fame man-
ſhall be ABC CA. Which was to be demonftrated.
PRO P. XIX. Fig. 23.
In every triangle ABC, under the greatest angle B is
fubtended the greateft fide AC.
For if AB be fuppofed equal to AC, then will
be the angle В (a) C, which is contrary to the
Hypothefis: And if ABAC, then ſhall be the angle
C(b) B, which is against the Hypothefis. Where-
fore rather AC AB; and after the fame manner
AC BC. Which was to be demonftrated.
PROP. XX. Fig. 24.
Of every triangle ABC, two fides BA, AC, any way
taken, are greater than the fide that remains BC,
Pro-
*
17. I.
a 3. 1.
b 5 1
c 16, 1.
d 9. ax.
a 5.1.
b 18. I.
14
The first Book of
a 3. 1.
b 5. 1.
c 9. ax.
d 19. I.
e conftr.
& 2. ax.
a 20. I.
b 4. ax.
c 16. 1.
a 3. 1.
b 3. poſt.
c 15. def.
d 1. ax.
a 1. poſt.
b 3. I.
C 22. I.
d 8. I.
Produce the line BA, (a) and take AD-AC, and draw
the line DC; (b) then ſhall the angle D be equal to
ACD; (c) therefore is the whole angle BCDD; (d)
therefore BD (e) (BA+AC) —BC. Which was to be
demonftrated.
PROP. XXI. Plate I. Fig. 25.
If from the utmoſt points of one fide BC, of a triangle
ABC, two right-lines BD, CD, be drawn to any point
within the triangle, then are both those two lines horter than
the two other fides of the triangle BA, CA; but contain a
greater angle, BDC.
Let BD be produced to E. Then is CE+ED (a)
CD, and BD common to both, (b) then ſhall be DB+
DE-ECCD+BD. Again, BA+AE (a) BE;
(6) therefore BA+AC— BE+EC. Wherefore 1. BAT
ACBD+DC. 2. The angle BDC (c) DEC (c)
A. Therefore the angle BDCA. Which was to be
demonftrated.
PROP. XXII. Fig. 26.
To make a triangle FKG of three right-lines FK, FG,
GK, which shall be equal to three right-lines given A,B,C.
Of which it is neceſſary that any two taken together be longer
than the third.
From the infinite line DE (a) take DF, FG, GH, equal
to the lines given A,B,C. Then if from the (b) centers
F and G at the diſtances of FD and GH, two circles be
drawn cutting each other in K, and the right-lines KF,
KG be joined, the triangle FKG fhall be made, (c) whoſe
fides FK, FG, GK, are equal to the three lines DF, FG,
GH, (d) that is, to the three lines given A, B, C. Which
was to be done.
PROP. XXIII. Fig. 27, and 28.
At a point A in a right-line given AH, to make a right-
lined angle A equal to a right-lined angle given D.
(a) Draw the right-line CF cutting the fides of the
angle given any ways; (b) make AG=CD; upon AG
(c) raiſe a triangle equilateral to the former CDF, fo
that AH be equal to DF, and GH to CF. then ſhall
you have the angle A (d) D. Which was to be done.
PROP.
EUCLIDE's Elements.
15
PROP. XXIV. Plate I. Fig. 29, 30.
If tavo triangles ABC, DEF have two fides of the one
triangle AB, AC, equal to two fides of the other triangle DE,
DF, each to each, and have the angle A greater than the an-
gle EDF contained under the equal right-lines, they shall have
alfo the Bafe BC greater than the bafe EF.
(a) Let the angle EDG be made equal to A, and
the fide DG (6)=DF (c) =AC; and let EG, and FG
be joined.
1. Cafe. If EG falls above EF; Becauſe AB (d)=DE,
and AC (e) DG, and the angle A (e) —EDG; (ƒ)
therefore is BC EG. But becauſe DF (e) DG; (g)
therefore is the angle DFG=DGF; (b) therefore is the
angle DFG EGF, and by confequence the angle
EFG, (b) EGF; (k) wherefore EG (BC) EF.
2. Cafe. If the bafe EF coincides with the bafe EG,
() it is evident that EG (BC) EF. Fig. 29, 31.
—
3. Cafe. If EG falls below EF, then becauſe DG+
GE (m) — DF-FE, if from both be taken away DG,
DF which are equal; EG (BC) remains (7) — EF.
Which was to be demonftrated. Fig. 29, 32.
PROP. XXV. Fig. 29, 30.
If two triangles ABC, DEF, have two fides AB, AC,
equal to two fides DE, DF, each to each, and have the
bafe BC greater than the bafe EF, they shall also have
the angle A contained under the equal right-lines greater
than the angle EDF.
For if the angle A be faid to be equal to EDF, (a)
then is the bafe BC-EF, which is againſt the Hypo-
thefis. If it be faid the angle A EDF, then (b)
will be BCEF, which is alſo againſt the Hypothefis.
Therefore AEDF. Which was to be demonftrated.
PROP. XXVI. Fig. 29, 31.
If two triangles BAC, EDG, have two angles of the one
B, C, equal to trvo angles of the other E, DGE, each to
his correfpondent angle, and have also one fide of the one equal
to one fide of the other, either that fide which lyeth betwixt the
equal angles, or that which is fubtended under one of the equal
angles; the other fides alfo of the one shall be equal to
the
a 23. 1.
b 3. 1.
c hyp.
d hyp.
e conftr.
f
4. I.
g 5. 1.
h 9. ax.
k 19. I.
19. ax.
m 21. 1.
n 5. ax.
a 4. I.
b 24. 1!
16
The firft Book of
a 3. 1.
b conftr.
chyp.
d 4. I.
e hyp.
fy. ax.
9.
g hyp.
h confir.
k 4. 1.
1 hyp.
m 16. 1,
a 16. 1.
a 15. 1.
b. 27. 1.
the other fides of the other, each to his correfpondent fide,
and the other angle of the one, shall be equal to the other
angle of the other.
1. Hypothefis. Let BC be equal to E G, which are
the fides that lie between the equal angles, Then I
fay BA ED, and AC=DG, and the angle A-EDG.
For if it be faid that EDBA, then (a) let EH be made
equal to BA, and let the line GH be drawn.
Becauſe AB(b) — HE, and BC (c) EG, and the an-
gle B()E, therefore fhall be the angle EGH (d)=
C(e)=DGE. (f)Which is abfurd, therefore AB ED.
After the fame manner AC may be proved equal to DG,
(a) then will the angle A be equal to EDG.
2. Hyp. Let A B be equal to DE, then I fay BC=
EG, and AC DG, and the angle A-EDG. For if
EG be greater than B C make EFB C, and join
DF. Now becaufe A B (g)=DE, and B C (b) = EF,
and the angle B (g) E; therefore will be the angle
EFD (k) == C(1)=EGD. (m) Which is abfurd. There-
fore is BC EG, and fo as before, AC DG, and
the angle ALDG. Which was to be dem.
PRO P. XXVII. Plate I. Fig. 33.
If a right-line EF; falling upon two right-lines AB, CD,
makes the alternate angles Ab F, Dc E, equal the one to the
other, then are the right-lines AB, CD, parallel.
If AB, CD be faid not to be parallel, produce them
till they meet in O, which being fuppofed, the outward
angle AF will be (a) greater than the inward angle
De E, to which it was equal by Hypothefis. Which
things are repugnant.
PRO P. XXVIII. Fig. 33.
If a right-line EF, falling upon two right-lines, AB,
CD, makes the outward angle AGE of the one line
equal to CH G the inward and oppofite angle of the
other on the fame fide, or make the inward angles on
the fame fide, AbÍ, СcG, equal to two right-angles,
then are the right-lines, AB, CD, parallel.
Hyp. 1. Becauſe by Hypothefis the angle Ab E
Cc G, (a) therefore are Bb H, Cc G, the alternate
angles equal; And (b) therefore are AB and CD paral-
lel.
Hyp
¦
EUCLIDE's Elements.
17
Hyp. 2. Becauſe by Hypothefis the angle AGH+
CHG to two right-angles,(a)—AGH+BGH,(†) there-
fore ſhall the angle CHG-BGH; and (c) therefore AB,
CD, are parallel, Which was to he demonfirated.
PRO P. XXIX. Plate I. Fig. 33.
If a right-line EF falls upon two parallels, AB, CD, it
will make both the alternate angles DHG, AGH, equal
each to the other, and the outward angle BGE equal to the
inward and oppofite angle on the fame fide DHG, as alfo
the inward angles on the fame fide AGH, CHG, equal to
two right-angles:
It is evident, that AGH+CHGtwo right-angles;
(a) otherwiſe AB, CD, would not be parallel, which is
contrary to the Hypothefis: But moreover the angle
DHG+CHG (6) two right-angles; therefore is DHG
(c):
= AGH (d) BGE. Which was to be dem:
Coroll. Fig: 7:
Hence it follows that every parallelogram AD having
one angle right A, the reft are alfo right.
For A+B (a) two right-angles. Therefore, where-
as A is right, (b) B muft be alfo right. By the fame
argument are C and D right-angles.
PRO P. XXX. Fig. 21.
Right-lines (ÁB, MN) parallel to one and the fame right-
line KL, are alſo parallel the one to the other.
Let CD cut the three right-lines given any ways.
Then becauſe AB, MN are parallel, the angle BII
will be (a) =LHI. Alfo becaufe KL and MN are
parallel, the angle LHI will be (a) MIC (b) There-
fore the angle BEI=MIC; (c) whence AB and MN
are parallel. Which was to be demonftrated.
PROP, XXXI. Fig. 34.
From a point given A, to draw a right-line AE, para!-
lel to a right-line given BC.
From the point A draw a right-line AD to any point
of the given right-line; with which at the point thereof
(a) A inake an angle DAE-ADC; (b) then will AE
and BC be parallel. Which was to be done.
PROP.
B
a 13. I.
b 3.ax.
C 27. I.
a 13. ax.
b 13.
I.
c 3 ax.
d 15. 1.
2 29. 1.
b 3. ax.
a 29. 1.
a 29. 1.
b 1. ax.
€ 27. 1.
a 23. 12
b 27. 1.
18
The first Book of
a 31. 1.
b 29. I,
C 2 ax.
d 19. ax.
e 13. I.
f 1. ax.
PROP. XXXII. Plate I. Fig. 35.
Of any triangle ABC one fide BC being drawn out, the
outward angle ACD shall be equal to the two inward op-
pofite angles A, B, and the three inward angles of the tri-
angle, A, B, ACB, ſhall be equal to two right-angles.
From C (a) draw CE parallel to BA. Then is the
angle A (b) ACE, and the angle B (b) —ECD.
Therefore A+B (c)≈ACE+ECD (d)=ACD. Which
was to be demonftrated.
I affirm ACD+ACB (e) two right-angles; (f)
therefore A+B+ACB two right-angles. Which was
to be demonftrated.
Coroll.
1. The three angles of any triangle taken together
are equal to the three angles of any other triangle taken
together. From whence it follows,
2. That if in one triangle, two angles (taken feve-
rally, or together) be equal to two angles of another tri-
angle (taken feverally, or together) then is the remaining
angle of the one equal to the remaining angle of the o-
ther. In like manner, if two triangles have one angle
of the one equal to one of the other, then is the
fum of the remaining angles of the one triangle equal to
the fum of the remaining angles of the other.
3. If one angle in a triangle be right, the other two
are equal to a right-angle. Likewife, that angle in a
triangle which is equal to the other two, is it ſelf a right-
angle.
4. When in an Ifofceles the angle made by the equal
fides is right, the other two upon the bafe are each of
them half a right-angle.
5. An angle of an equilateral triangle makes two third.
parts of a right-angle. For one third of two right-angles
is equal to two thirds of one.
Schol.
By the help of this propofition you may know how
many right-angles the inward and outward angles of a
right-lined figure make; as may appear by thefe two
following Theorems.
THE O.
EUCLIDE's Elements.
19
THEOREM I.
All the angles of a right-lined figure do together make
wice as many right-angles, abating four, as there are fides
of the figure.
From any point within the figure let right-lines
be drawn to all the angles of the figure, which ſhall
reſolve the figure into as many triangles as there are
fides of the figure. Wherefore, whereas every triangle
affords two right-angles, all the triangles taken together
will make up twice as many right-angles as there are
fides. But the angles about the faid point within
the figure make up four right; therefore, if from
the angles of all the triangles you take away the
angles which are about the faid point, the remaining
angles, which make up the angles of the figure, will
make twice as many right-angles, abating four, as
there are fides of the figure. Which was to be demon-
ftrated.
Coroll.
Hence all right-lined figures of the fame fpecies have
the fums of their angles equal.
THEOREM II.
All the outward angles of any right-lined figure, taken to-
gether, make up four right-angles.
For every inward angle of a figure, with the out-
ward angle of the fame, make two right-angles; there-
fore all the inward angles, together with all the out-
ward, make twice as many right-angles as there are
fides of the figure; but (as has been just fhewn) all the
inward angles, with four right, make twice as many
right as there as fides of the figure, therefore the out-
ward angles are equal to four right-angles. Which was
to be demonfirated.
Coro it.
All right-lined figures, of whatfoever fpecies, have
the fums of their outward angles equal.
B 2
PROP.
20
The first Book of
a 29. 1.
b 4. I.
C 27. I.
a hyp.
b
29. I.
C 2. ax.
d 26. 1.
a 27. 1.
PROP. XXXIII. Plate I. Fig. 7.
If two equal and parallel lines AB, CD, be joyned to-
gether with two other right-lines, AC, BD, then are
thofe lines alfo equal and parallel.
Draw a line from C to B. Now becauſe AB and CD
are parallel, and the angle ABC (a) = BCD; and alſo
by hypothefis A B≈CD, and the fide C B common,
therefore is AC (b) BD, and the angle ACB (b) =
DBC (c) whence alfo AC, BD, are parallel,
PRO P. XXXIV, Fig. 7.
In parallelograms, as ABDC, the oppofite fides AB, CD,
and AC, BD, are equal each to the other; and the oppofite
angles A, D, and ABD, ACD, are alſo equal; and the
diameter BC bijects the fame.
Becauſe AB, CD, (a) are parallel, (b) therefore is the
angle ABC BCD. Alfo becauſe AC, BD, are (a) pa-
rallel, (b) therefore is the angle ACB=CBD; (c) there-
fore the whole angle ACD ABD. After the fame
manner is AD. Moreover becauſe the angles ABC,
ACB, lie at each end of the fide CB, and are equal to
BCD, CBD, (d) therefore is AC
CD, and fo the triangle ABC
be demonfirated.
Schol.
BD, and AB‍(d)=
CBD. Which was to
Every four-fided figure ABDC, having the oppofite fides
equal, is a parallelogram.
For by 8. 1. the angle ABC BCD; (a) wherefore
AB, CD, are parallel. In like manner is the angle BCA
CBD; (a) wherefore AC, BD, are alfo parallel, (b)
b 35. def. 1. Therefore ABCD is a parallelogram. Which was to be
Fig. 36.
16.36.
demonftrated.
From hence we may more expeditiouſly draw a
parallel CG to a right-line given, AB, thro' a point
affigned, C.
Take in the line AB any point, as E. From the
centers E and C at any diſtance draw two equal circles
EF, CD. From the center F with the diftance EC
draw a circle FD, which fhall cut the former circle
CD in the point D. Then fhall the line drawn CG
be parallel to AB, for, as it was before demonſtrated,
CEFD is a parallelogram.
PROP.
1
EUCLIDE's Elements.
21
PRO P. XXXV. Plate I. Fig. 37. .
Parallelograms, BCDA, BCFE, which fand upon
the fame baſe BC, and between the fame parallels AF,
BC, are equal one to the other.
:)
For AD (a) BC (a) EF, add DE common to
both; (b) then is A E D F. But alfo AB («t
DC, and the angle A (c) =CDF. (d) Therefore is
the triangle ABE-DCF. Take away Dg E common
to both triangles; (e) then is the Trapezium AB g D =
CF; add B g C, common to both; (f) then is the
parallelogram ABCD EBCF. Which was to be demon-
frated.
E
The demonftration of any other cafes, is not uniike,
but much more plain and eaſy.
Schol. Fig. 7.
{
If the fide AB, of a right-angled parallelogram ABCD,
be conceived to be carried along perpendicularly thro'
the whole line AC, or AC thro' the whole line AB, the
area or content of the rectangle ABCD fhall be pro-
duc'd by that motion. Hence a rectangle is faid to
be made by the drawing or multiplication of two
contiguous fides. For example; let AB be ſuppoſed
four foot, and AC three; draw three into four, there
will be produced twelve fquare feet for the area of
the rectangle.
This being fuppofed, the dimenfion of any parallelq-
gram (EBCF) is found out by this theorem. For the
area thereof is produced from the altitude BA drawn
into the bafe BC. For the area of the rectangle AC
parallelogram ERCF, is made by the drawing of
BA into BC, therefore, &c.
PROP. XXXVI. Fig 37.
Parallelograms BCDA, GHF E, ſtanding upon equal
bafes BC, GH, and betwixt the fame parallels AF, BH,
are equal one to the other.
Draw BE and CF. Becauſe BC (a) = GH(b) =
EF, (c) therefore is BCFE a parallelogram.
the parallelogram BCDA (d) BCFE (1)
Which was to be demonftrated.
B 3
f-
a 34. 1.
b 2. ax.
C 29. I.
d
4. I.
e 3. ax.
£ 2. ax.
Fig. 37.
a hyp.
b 34. 1.
Whence
GHFE.
C 33. I.
d
35. 4.
PRQ P.
22
The first Book of
a 31. 1.
b 34. I.
€ 35. I.
and 7. ax.
a 34. 1.
b 36. 1.
an! 7. ax.
€ 34. I.
a 37. I.
b hyp•
c 9. ax.
a 38. 1.
b hyp•
e J. ax.
PROP. XXXVII. Plate I. Fig. 38.
Triangles, BCA, BCD, ftanding upon the fame bafe
BC, and between the fame parallels BC, EF, are equal
the one to the other.
(a) Draw BE parallel to CA, (a) and C F parallel
to B D. Then is the triangle BCA (6) half Pgr.
ВСАЕ (c) half BDFC (6) ≈ BCD. Which was to
be demonftrated.
1
PROP. XXXVIII. Fig. 38.
Triangles, BCA, DFC, fet upon equal baſes BC, DF,
and between the fame parallels EF, BC, are equal the
one to the other.
Draw EB parallel to AC, and DB parallel to FC.
Then is the triangle BCA (a) half Pgr. BCAE (6)
half BCDF (©) — CLF.
CDF. Which was to be demon-
Strated.
Schol.
If the bafe BC be greater than DF, then is the
triangle BACDFC, and fo on the contrary.
PRO P. XXXIX. Fig. 39.
Equal triangles BCA, BCD, ſtanding on the fame baſe
BC, and on the fame fide are alfo between the fame paral-
·lels AD, BC.
If you deny it, let another line AF be parallel to
BC; and let CF be drawn. Then is the triangle CBF
(a)= CBA (b) = CBD. (c) Which is abfurd.
PROP. XL. Fig. 40.
Equal triangles BCA, CFD, ſtanding upon equal bafes
BC, CF, and on the fame fide, are betwixt the fame
parallels.
If you deny it, let another line AH be parallel to
BF, and let FH be drawn. Then is the triangle
CFH (a) BCA (b)=CFD. (c) Which is abfurd.
PRO P. XLI. Fig. 38.
If a Pgr. AEBC have the fame bafe BC with the triangle
BCE, and be between the fame parallels DE, BC, then is the
Pgr. AEBC double to the triangle BCE.
Let
EUCLIDE'S Elements.
23
Let the line AC be drawn. Then is the triangle BCA
(a) —BCE; therefore is the Pgr. ABCD (6) =2BCA
(e) zBCE. Which was to be demonftrated.
Schol.
From hence may the area of any triangle BCE be
found, for whereas the area of the Pgr. ABCD is produ-
ced by the altitude drawn into the baſe, therefore ſhall
the area of a triangle be produced by half the altitude
drawn into the bafe, or half the bafe drawn into the
altitude; thus, if the baſe BC be 8, and the altitude 7,
then is the area of the triangle BCE 28.
PROP. XLII. Plate I. Fig. 41.
To make a Pgr. ECGF equal to a triangle given ABC in
an angle equal to a right-lined angle given.
Through A (a) draw AG parallel to BC, (b) make the
angle BCG the angle given; (c) bifect the baſe BC in
E, and draw EF parallel to CG; then is the problem
refolved.
For (AE being drawn) the angle ECG is equal to the
given angle by conftruction, and the triangle BAC (d)
=2 AEC (c)= Pgr. ECGF. Which was to be done.
PROP. XLIII. Fig. 9.
In every Pgr. AGEL, the complements LD, GD, of thoſe
Pgrs. CK, KF, which fand about the diameter, are equal
one to the other.
For the triangle AEL (a)=AEG, and the triangle
ADC (a) =ADB; and the triangle DEK (a) -DEF (6)
Therefore the Pgr. LDDG. Which was to be demon-
ftrated.
PRO P. XLIV. Fig. 42.
To a given right-line A, to apply a parallelogram FL,
equal to a given triangle LBC, in a given angle C.
(a) Make a Pgr. FD equal to the triangle LBC, fo that
the angle GFE may be equal to C. Produce GF till
FH be equal to the line given A. Through H (b) draw
IL parallel to EF, which let DE produced meet in I,
let DG produced meet with a right-line drawn from I
through F in the point K; thro' K (6) draw KL parallel
to GH, which let EF continued meet at M, and IH
at L. Then fhall FL be the Pgr. required.
B 4
For
a 37. I.
b 34.1.
c 6. ax.
a 31.
b
1.
23. I.
C IO. 1.
d 38. 1.
e 41. I.
a 34. 1.
b
3 ax.
a 42. I.
b 31. 1.
24
The first Book of
CI
d 15. 1.
2 44. I.
b 19. ax.
c conftr.
2 11. I.
I.
b 3.
c confir.
d 28. i.
e conſtr:
33.
f
I.
g cor. 29. 1.
h 29. dif.
a 12. ar.
b 29. def.
€ 4. I.
4 41. 1.
For the Pgr. FL (c) =FD= LBC; (d) and the angle
MFH-GFÈ=C. Which was to be done.
PROP. XLV. Plate I. Fig. 43, and 44.
Upon a right-line given FG, and in a given angle A, to
make a Pgr. F L, equal to a right-lined figure given
ABCD.
Refolve the right-lined figure given into two triangles
BAD, BCD, then (a) make a Pgr. FH-BAD, fo that
the angle F may be equal to A. FI being produced,
(a) make on HI the Pgr. ILBCD. Then is the Pgr.
FL (6)=FH+-IL (c)=ABCD. Which was to be done.
Schol.
Hence is eafily found the excefs, whereby any right-
lined figure exceeds a leſs right-lined figure.
PROP. XLVI. Fig. 6.
Upon a right-line given CD to defcribe a fquare DB.
(a) Erect two perpendiculars CB, DA, (6) equal to the
line given CD; then join BA, and the thing reqired is
done.
For, whereas the Angle C+D (c) two right-an-
gles (d) therefore are AD, BC parallel. But they are
alfo (e) equal; (f) therefore AB, DC are both parallel
and equal; therefore the figure AC is a Pgr. and equila-
teral. Moreover the angles are all right, (g) becauſe one
A, is right; (b) therefore DB is a fquare. Which was
to be done.
After the fame manner you may eaſily deſcribe a rect-
angle contained under two right-lines given.
PRO P. XLVII. Fig. 45.
In right-angled triangles BAC, the fquare BE, which is
made on the fide BC that fubtends the right-angle BAC, is
equal to both the ſquares BG, CH, which are made in the
fides AB, AC, containing the right-angle.
Join AE, and AD; and draw AM parallel to CE.
Becaufe the angle DBC (a)=FBA, add the augle ABC
common to them both; then is the angle ABD— FBC.
Morcover, AB (6)=FB, and BD (b) =BC; (c) there-
fore is the triangle ABD=FBC. But the Pgr. BM (d)
2 ABD,
}
EUCLIDE's Elements.
25.
d 41. 1.
e 6. ax.
2 ABD, and the Pgr. (d) BG= 2FBC (for GAC is one
right-line, by Hypothefis, and 14. 1.) (e) therefore is the
Pgr, BM-BG. By the fame way of argument is the
Pgr. CM CH. Therefore is the whole BE (ƒ)BG+ £ 2. ax.
CH. Which was to be demonftrated.
Schol.
This most excellent and uſeful theorem hath defer-
ved the title of Pythagoras his theorem, becauſe he was
the inventor of it. By the help of which the addition
and ſubtraction of fquares are performed; to which pur-
poſe ſerve the two following problems,
PROBLEM I. Plate I. Fig. 46.
To make one ſquare equal to any number of ſquares given.
Let three fquares be given, whereof the fides are
AB, BC, CE. (a) Make the right-angle FBZ, having
the fides infinite; and on them transfer AB and BC;
join AC, then is ACq (b) ABq BCq. Then trans-
fer AC from B to X, and CE the third fide given
from B to E; join EX. (b) Then is EXq EBq
(CEq) + BXq (ACq)(c) CEq + ABq + BCq.
Which was to be demonstrated.
PROBLEM II. Fig. 2.
Two unequal right-lines being given AE, EG, to make
a fquare equal to the difference of the trvo fquares of the
given lines AE, EG.
From the center E, at the diſtance of AE, deſcribe
a circle, and from the point G erect a perpendicular
FG, meeting the circumference in F; and draw EF.
(a) Then is EFq (EAq) = EGq + FGq, (b) There-
EAqEGq=GFq. Which was to be done.
PROBLEM III. Fig. 45.
Any two fides of a right-angled triangle ABC, being
known, to find out the third.
Let the fides AB, AC, encompaffing the right-angle,
be, the one 6 foot, the other 8. Therefore, whereas
ACq - ABq 64+ 36 100 BCq, thence is BC
= √100 = 10,
Andr.
Tacq.
a 11. 1.
II.
b 47. I.
c 2. ax.
a 47. 1.
b 3. Qx.
47. !
Now
26
The first Book of
47. 1.
a 47. 1.
** See the
following
theor.
b 8. 1.
€ conftr.
2 34. I.
4. I.
b
6. ax.
a 46. 1.
&
b 1. part,
c byp.
d 9. ax.
Now, let the fides AB, BC, be known, the one 6
Therefore fince BCq A Bq
ACq, whence AC=√ 64
foot, the other 10,
100 36 64
Which was to be done.
PRO P. XLVIII. Plate I. Fig. 47.
8.
If the fquare made upon one fide BC of a triangle be
equal to the fquares made on the other fides of the tri-
angle AB, AC, then the angle BAC comprehended under
the two other fides of the triangle AB, AC, is a right-
angle.
Perpendicular to AC draw AD = AB, and join CD.
Now is (a) CDq= ADq+ ACq=ABq-+ ACq=
BCq. * Therefore is CD BC. And therefore the
triangles CAB, CAD, are mutually equilateral. Where-
fore the angle CAB (6) CAD (c) a right-angle.
Which was to be demonftrated.
=
Schol.
We affumed in the demonſtration of the laft Propo-
fition, CD = BC, becaufe CDq was equal to BCq:
Our affumption we prove by the following
THE ORE M. Fig. 48.
The fquares AF, CG of equal right-lines AB, CD, are
equal one to the other: And the fides AB, CD, of equal
Squares AF, CG, are equal one to the other.
1. Hypothefis. Draw the diameters EB, HD. Then
it is evident that AF is (a) equal to the triangle EAB
twice taken, and (b) equal to the triangle HCD twice
taken, and equal to (a) CG. Which was to be demonſtrated.
2. Hyp. If it may be, let CD be greater than A B.
Make CT AB, and let CS CTq. `Therefore is
CS (b) = EB (c) = CG. Which is abfurd.
Corol
After the fame manner any rectangles equilateral one
to another, are demonftrated to be alfo equal.
The End of the first Book.
The
!
[ 27 ]
The SECOND BOOK
O F
EUCLIDE's
ELEMENT S.
*
I.
E
Definitions.
Very right-angled Parallelogram ABCD, is
faid to be contained under two right-lines
AB, AC, comprehending a right-angle.
Therefore when you meet with fuch as thefe, the rectan-
gle under BA, AC, or more briefly the rectangle BAC,
or BAX AC (or ZA, for Z x A) the rectangle meant
is that which is contained under the right-lines BA, AC,
fet at right-angles.
II. In every Pgr. ALGE, any one of thofe paral-
lelograms which are about the diameter, together with
its two complements is called a Gnomon. As the Pgr.
FB DL FK (BEC) is a Gnomon; and likewiſe the
Pgr. FB, DL - BC (FAK) is a Gnomon,
PRO P. I. Fig. 49.
If two right-lines AF, AB, are given, and one of them
AB be divided into as many parts or fegments as you please;
the rectangle comprehended under the two whole right-lines
AB, AF, ſhall be equal to all the rectangles contained
under the whole line AF, and the feveral fegments, AD,
DE, EB.
(a) Set AF perpendicular to AB. Thro' F (a) draw
an infinite line FG perpendicular to AF. From the points
D, E, B, erect perpendiculars DH, EI, BG. Then
is AG a rectangle comprehended under AF, AB, and
is (6) equal to the rectangles AH, DI, EG, that is (be-
caufe DH, EI, AF, (c) are equal) to the rectangles under
AF, AD, under AF, DE, under AF, EB. Which was
to be demonftrated.
Schol.
Plate I.
Fig. 7.
Fig. 9.
a II. 1.
I
b 19
19 ax 1.
C 34, I.
28
The Second Book of
a 1. 2.
21.
b 2. ax.
*
*
Ap
I. 2,
19, ax.
Schol.
If two right-lines given are both divided into how many
parts foever, one whole multiplied into the other hall bring
out the fame product, as the parts of one multiplied into the
parts of the other.
DA+
For let Z be = A + B + C, and Y DE; then,
becauſe 1Z (a) = DA + DB + DC, and EZ (a)=
EAEBEC, and YZ (a)= DZEZ, (b) There-
fore ZY will be=DA + DB + DC + EA+ EB +
EC. Which was to be demonftrated.
From hence we have a method of multiplying compound
lines into compound ones. For if the rectangles of all the
parts be taken, their fum fhall be equal to the rectangle
of the wholes.
But whenfoever in the multiplication of lines into
themſelves you meet with thefe figns-intermingled with
thefe, you muſt alſo have particular regard to the
figns. For of multiplied into arifeth ; but of-
into arifeth ex. gr. let A be multiplied into
B-C; then becauſe A is not affirmed of all B, but
only of that part of it, whereby it exceeds C, therefore
AC muſt remain denied; fo that the product will be AB
- AC. Or thus; becauſe B confifts of the parts C and
B-C,* thence ABACAX BC, take away
AC from both, then AB → AC=AXB-C. In like
manner, if A be to be multiplied into B C, then
fince by virtue of the fign A is not denied of all R,
but only of fo much as it exceeds C, therefore AC muſt
remain affirmative, whence the product will be AB
AC. Or thus; becauſe AB ACTA × B➡C;
C ; take
away all from both fides, and there will be AB
АС
AC — AXB - C; add AC to both, and it will be
ABACE AX B-C.
This being fufficiently underſtood, the nine following
propofitions, and innumerable others of that kind,
arifing from the comparing of lines multiplied into
themſelves (which you may find done to your hand in
Vieta, and other analytical Writers) are demonftrated
with great facility, by reducing the matter for the moſt
part to almoſt a fimple work.
*
Furthermore, it appears that the product arifing
from the multiplication of any magnitude into the parts
of any number is equal to the product arifing from
the multiplication of the fame into the whole number:
As 5A +7A= 12 A, and 4 A XSA+4 Ax7A==
4 A
EUCLIDE's Elements,
29
4
AX 12 A. Wherefore what is here delivered of the
multiplying of right-lines into themſelves, the fame may
be underſtood of the multiplying of numbers into them-
felves, ſo that whatſoever is affirmed concerning lines
in the nine following Theorems, holds good alſo in
numbers; feeing they all immediately depend on, and
are deriv'd from this firſt.
PROP. II. Plate I. Fig. 50.
If
a right-line Z be divided any wife into two parts, the
rectangles comprehended under the whole line Z, and each of
the fegments A, E, are equal to the fquare made of the whole
line Z.
I fay that ZA+ZE=Zq. For take BZ; (a)
then is BABE BZ, that is (becauſe BZ) ZÁ
+ZE Zq. Which was to be demonftrated.
PRO P. III. Fig. 50.
If a right-line Z, be divided any wife into two parts,
the rectangle comprehended under the whole line Z, and one
of the fegments E, is equal to the rectangle made of the
Jegments A, E, and the fquare defcribed on the ſaid ſegment E.
I ſay ZE = AE + Eq. (a) For EZ= EA + Eq.
PRO P. IV. Fig. 50.
If a right-line Z be cut any wife into two parts, the fquare
defcribed on the whole line Z, is equal to the fquares deſcribed
on the fegments A, E, and to twice a rectangle made of the
Segments A, E, taken together.
I fay that Zq Aq +Eq-+- 2AE. For ZA (a)=Aq
+AE, and ZE (a) EqEA. Therefore whereas
ZA + ZE (6)=Zq, (c) thence is Zq Aq+ Eq +
2AE. Which was to be demonftrated.
Otherwife thus; Upon the right-line AB make the
ſquare AD, and draw the diameter EB; thro' C, the
point wherein the line AB is divided, draw the perpen-
dicular CF; and thro' the point G draw HI paral-
lel to AB.
Becauſe the angle EHGA is a right-angle, and
AEB is (d) half a right (e) therefore is the remaining
angle HGE half a right-angle. Therefore is HE (ƒ)=
HG (g) — EF (g) — AC, ſo that HF (b) is the ſquare of
the right-line AC. After the fame manner is CE proved
fo
a 1. z.
a 1. 2.
a 3. 2.
b. 2. 2.
C I, ax.
Fig. 51.
d 4. cor.
32. I.
e 32. I.
f 6. 1.
g 34. I.
7
h 29. def.
to
30
The Second Book of
to be CBq. Therefore AG, GD, are rectangles under
k 19. ax. 1. AC, CB, wherefore the whole fquare AI) (®) = A€q
+CBq+2ACB. Which was to be demonftrated.
a 4. I.
b 3. 2.
c hyp.
d 1. 2.
Sch. 1. 2.
a 5.2. &
3. ax.
b 4.2.
c3. ax.
Coroll.
1. Hence it appears that the Pgrs. which are about
the diameter of a fquare are alſo ſquares themſelves.
2. That the diameter of any fquare bifects it angles.
3. That if a Z, then is Zq 4 Aq, and Aq
Žq. As on the contrary, if Zq= 4 Aq, then is A
12.
Z.
A-
1-
PRÓ P. V.
1B
จ
-B If a right-line AB be cut in-
C D to equal parts AC, CB, and in-
to unequal parts AD, DB, the rectangle comprehending under
the unequal parts AD, DB, together with the fquare that is
made of the difference of the parts CD, is equal to the
Square defcribed on the half line CB.
I fay that CBq ADB + CDq.
CBq.
Sia)
For thefe are (a) CDq + CDB + DBq + CDB.
all equal. CDq+(6) CBD (c) (ACX BD)+CDB.
(CDq+(d) ADB.
This theorem is fomewhat differently exprefs'd and
more eafily demonftrated thus; A rectangle made of the
fum and the difference of two right-lines A, E, is equal to
the difference of the Squares of thoſe lines.
For if AE be multiplied into A-E, there arifeth
Aq-AE+EA-Eq=Aq-Eq. Which was to be de-
Schol. Plate I. Fig. 52.
monftrated.
If the line AB be divided otherwiſe, (viz.) nearer to
the point of biſection, in E; then is AEB ADB.
For AEB (a) =CBq-CEq, and ADB (a) = CBq –
CDq. Therefore, whereas CDq-CEq, thence is AEB
ADB. Which was to be demonftrated.
Coroll.
1. Hence is ADq+DBq C¯ AEq+EBq. For ADq
+DBq+2 ADB (6) — ABq (b) — AÉq†ÊВq+2AEB.
Therefore becaufe 2AEB 2ADB, fhall ADq f⋅ D Bq •
AEq+EBq. Which was to be demonftrated.
2. Hence is ADq †-DBq—AEq (c) —EBq=2AEB—
2 ADB.
PKOP.
EUCLID E's Elements.
31
PRO P. VI. Plate I. Fig. 53.
If a right-line A be divided into two equal parts, and ano-
ther right-line E, added to the fame directly in one right-line,
then the rectangle comprehended under the whole and the line
added, (viz. A+E,) and the line added E, together with the
Square which is made of the line A, is equal to the fquare
of A+E taken as one line.
I
4
I
2
I fay that Aq. ( (a) Q. & A) +AE+Eq=Q.
A+E. (a) For, Q. ATE=Aq+Eq AE. Which
was to be demonftrated.
¦
Coroll.
Hence it follows, that if 3 right-lines E, EA, E†
A be in arithmetical proportion, then the rectangle
contained under the extreme terms E, EA, together
with the fquare of the difference A, is equal to the
ſquare of the middle term E+A.
PROP. VII. Fig. 50.
If a right-line Z be divided any wife into two parts, the
Square of the whole line Z, together with fquare made of
one of the fegments E, is equal to a double rectangle compre-
hended under the whole line Z, and the faid fegment E, to-
gether with the fquare made of the other fegment A.
I fay, that Zq+Eq=2ZE+Aq. For Zq (a) Aq+
Eq-1-2AE, and 2 ZE (6)—2 Eq+2AE. Which avas to
be demonfirated.
Coroll.
Hence it follows, that the fquare of the difference of
any two lines Z, E, is equal to the fquare of both the
lines lefs by a double rectangle comprehended under the
faid lines.
(c) For Zq+Eq-2ZE—Aq—Q. Z-E.
PRO P. VIII. Fig. 50.
If a right-line Z be divided any wife into two parts, the
rectangle comprehended under the whole line Z, and one of
the fegments E four times, together with the square of the
other fegment A, is equal to the fquare of the whole line
Z, and the fegment E, taken as one line ZE.
I fay, that 4ZE+Aq=QZ +E. For 2 ZE (0)=
Zq+Eq Aq. Therefore 4 ZE + AqZq+Eq
+2ZE (6)=QZE. Which was to be demonftrated.
—
+
PROP.
a4 & 3.
Cor. 4. 2.
a 4. 2.
b 3.2.
C 7. 2. and
3. ax.
a 7. 2. and
3. ax.
b 4. 2.
32
The Second Book of
a 4.2.
b byp.
€ 7. 2.
d 2. ax.
a 4, 2.
·
PROP. IX. Plate I. Fig. 54.
If a right-line AB be divided into equal parts AC, CB
and into unequal parts AD, DB, then are the fquares of the
unequal parts AD, DB, together, double to the fquare of the
half line AC, and to the fquare of the difference CD.
I fay that ADq+DBq=2 ÁCq +2CDq. For ADq
+DBq (a) = ACq + CDq + 2 ACD+DBq. But
2ACD (b) (2BCD)+DBq () CBq (ACq) + CDq.
(d) Therefore ADq + DBq 2 ACq+2CDq. Which
was to be demonftrated.
This may be otherwiſe delivered and more eafily de-
monſtrated thus; the aggregate of the fquares made of the
fum and the difference of two right-lines A, E, is equal
to the double of the ſquares made from thofe lines.
For QA+E (a) Aq+ Eq+2AE, and Q. A-
b cor. 7. 2. E(6)=Aq+Eq-2AE. Theſe added tegether make 2Aq
+2Eq. Which was to be demonftrated.
a 4. 2.
b cor. 4. 2.
€ 4.2.
a 46. 1.
b 10. 1.
PROP. X. Fig. 53.
If a right-line A be divided into trvo equal parts, and ano-
ther line be added in a right-line with the fame, then is the
Square of the whole line together with the added line (as
being one line) together with the fquare of the added line
E, double to the fquare of half Ä, and the added line E,
taken as one line.
I fay that Eq+Q.A + E, ¿. e. (a) Aq† 2 Eq42AE
= 2 Q₂ = A + 2 QAE, For 2 QA (6) Aq.
And 2 QA+E (c) = ½ Aq + 2 Eq + 2AE. Which
was to be demonftrated,
PROP. XI. Fig. 55.
To cut a right-line given AB, in a point G, ſo that the
rectangle comprehended under the whole line AB, and one
of the fegments BG, ſhall be equal to the Square, that is made
of the other Segment AG.
Upon AB (a) deſcribe the ſquare AC (b) Bifect the fide
AD in E, and draw the line EB; from the line EA
produced take EFEB. On AF make the ſquare AH.
Then is AH AB× BG.
For
¡
EUCLIDE's Elements.
33
For HG being drawn out to I; the rectangle DH-+-
EAq (c)=EFq (d) —EBq (e) —BAq+EAq: Thereforec 6. 2.
is DH (ƒ)=BAq to the fquare AC. Take away AI
d conftr.
common to both, there remains the fquare AH=GC,
€ 47. 1.
that is, AGqABX BG. Which was to be done.
£ 3. ax.
Schol.
This propofition cannot be performed by numbers;
* for there is no number that can be fo divided, that
the product of the whole into one part fhall be equal to
the fquare of the other part.
PROP. XII. Plate I. Fig. 56.
In obtufe-angled triangles as ABC, the fquare that is made
of the fide AC, fubtending the obtufe angle ABC, is greater
than the fquares of the fides BC, AB, that contain the obtufe
angle ABC, by a double rectangle contained under one of the
fides BC, which are about the obtufe angle ABC, on which
fide produced the perpendicular AD falls, and under the
Line BD, taken without the triangle from the point on which
the perpendicular AD falls to the obtufe angle ABC.
I fay that ACq=CBq+ABq+2CBX BD.
For theſe are
all equal
ACq.
(a) CDq+ADq.
(b) CBq+2CBD+BDq+ADq;
CBq+2CBD+ (c) ABq.
Scholium.
Hence, the fides of any obtufe-angled triangle ABC being
known, the fegment BD intercepted betwixt the perpendicular
AD, and the obtuſe angle ABC, as also the perpendicular
it felf AD, ſhall be easily found out.
Thus, Let AC be 10, AB 7, CB 5. Then is ACq 100,
ABq 49, CBq 25. And ABq+CBq=74. Take that
out of 100, then will 26 remain for 2 CBD. Where-
fore CBD fhall be 13; divide this by CB 5, there will
2 be found for BD. Whence AD will be found out
by the 47. 1.
PRO P. XIII. Fig. 57.
In acute-angled triangles as ABC, the fquare made of the fide
AB, fubtending the acute angle ACB, is less than the fquares
angle
C
made
f
#6. 13,
a 47. 1.
b 4. 2.
€ 47. I.
34
The fecond Book of
a 47. I.
b 7. 2.
€ 47. I.
a 45. I.
b 10. I.
* 46. 1.
c conftr.
d 5. 2.
2. and
3. ax.
e 47. 1. and
3. ax.
made of the fides AC, CB, comprehending the acute angle
ACB, by a double rectangle contained under one of the fides
BC, which are about the acute angle ACB, on which the
perpendicular AD falls, and under the line DC, taken with-
in the triangle from the perpendicular AD, to the acute.
angle ACB.
I fay that ACq+BCq=ABq+2BCD.
AСq+BCq.
For theſe are (a) ADq+DCq+BCq.
equal.) ADq+BDq+2BCD.
(c) ABq+2BCD.
Coroll.
Hence, the fides of an acute-angled triangle ABC being
known, you may find out the fegment DC, intercepted betwixt
the perpendicular AD, and the acute angle ACB, as also the
perpendicular it felf AD.
Let AB be 13, AC 15, BC 14. Take ABq (169) from
ACq+BCq, that is, from 225-196421. Then re-
mains 252 for 2BCD, wherefore BCD will be 126, di-
vide this by BC 14, then will 9 be found out for DC.
From whence it follows, AD√ 225—81—12.
PROP. XIV. Plate I. Fig. 58.
To find a fquare MC equal to a right-lined figure given A.
(a) Make the rectangle DBA, and produce the great-
er fide thereof DC to F, fo that CF CB, (b) biſect DF,
in G, about which as the center, at the diſtance of GF,
defcribe the circle FHD, and draw out BC, till it meets
the circumference in H. Then fhall be CHq=*MC
=A.
For let GH be drawn.
DCF (d) =GFq-GCq (e)
was to be done.
Then is A (c) =DB (c) =
=HCq (c) MC. Which
The End of the fecond Boo K.
The
1
Plate I. Facing Pag. 34.
Fig. 1
1.
D C
A-
Fig. 9·B G
B
F
Fig. 2
BA
C
G
E G
A
Fig. 3. Fig. 4.
B
F
B
Fig. 5.
Fig.
12
C D
Fig.6.
D
H
F
Fig. 10.
A
EF
Fig. 11.
睿
​I
KE
Fig. 15.
N
B-
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Fig. 16
G
Fig.17
D
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R
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B
A
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Fig. 8,
D
D
D
B
Fig. 18.
B
Fig. 13.
DA
Fig. 14.
B
E C
D
A
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Fig.20.
B
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A
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Fig. 21.
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Fig.22
Fig.19.
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Fig. 25.
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Fig. 29.
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Fig.31.
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Fig.35
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Fig.51
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A
·M
[ 35 ]
The THIRD BOOK
O F
EUCLIDE's
ELEMEN T S
I.
E
Definitions.
Qual circles (GABC, HDEF) are fuch whoſe
diameters are equal; or, from whoſe centers
right-lines drawn GA, HD, are equal.
II. A right-line MO, is faid to touch a circle GABC,
when touching the fame, and being produced, it cutteth
it not.
The right-line GCO cuts the circle GABC.
III. Circles are faid to touch one the other, which
touch, but cut not one the other. As Fig. 8.
Thofe of Fig. 7. cut each other.
IV. In a circle GABD, right-lines FE, KL, are faid
Plate II.
Fig. 1, 2.
Fig. 2.
to be equally diſtant from the center, when perpendiculars Fig. 3.
GH, GN, drawn from the center G to them, are equal.
And that line BC is faid to be furtheft diftant from it, on
which the greater perpendicular GI falls.
V. A fegment of a circle (BPC) is a figure contained
under a right-line BC, and a portion of the circumfe-
rence of a circle BPC.
VI. An angle of a fegment CBP, is that angle which
is contained under a right-line BC, and the circumfe-
rence of a Circle PB.
VII. An angle BPC is faid to be in a fegment BPC,
when in the circumference thereof fome point P is taken,
and from it right-lines BP, CP, drawn to the ends of
the right-line BC, which is the bafe of the fegment;
then the angle BPC contained under the adjoined lines
BP, CP, is faid to be an angle in a fegment.
VIII. But when the right-lines BP, CP, comprehend-
ing the angle BPC, receive any periphery of the circle
BAC, then the angle UPC is faid to stand upon that
periphery.
C 2
IX. A
Fig. 2.
;
36
The third Book of
Fig. 1.
Fig. 1, 2,
a 15. def. 1.
b 8. 1.
c 10. def. 1.
d 12. ax.
e 9. ax.
Fig. 5.
Andr.
Tacq.
IX. A fector of a circle (HRS) is when an angle RHS
is fet at the center H of that circle; namely, that figure
KHS comprehended under the right-lines RH, SH, con-
taining the angle, and the part of the circumference RS,
intercepted between them.
X. Like fegments of a circle (ABC, DFE) are thoſe
which include equal angles (ABC, DEF;) or, in
which the angles ABC, DEF, are equal.
PROP. I. Plate II. Fig. 4.
To find the center F of a given circle ABC.
Draw a right-line AC any-wife in the circle, which
bifect in E; thro' E draw a perpendicular DB, and biſect
the fame in F; the point F fhall be the center.
If you deny it, let G, a point without the line BD, be
the center (for it cannot be in the line BD, fince that is
divided unequally in every point but F) let the lines
GA, GC, GE, be drawn. Now if G be the center, (a)
then is GA-GC, and AE-EC, by conftruction, and
the fide GE common. (b) Therefore are the angles GEA,
GEC, equal, and (c) confequently right. (d) Therefore
the angle GEC=FEG. (e) Which is abfurd.
Coroll.
Hence, if in a circle a right-line BD bifect any right-
line AC at right-angles, the center ſhall be in the cut-
ting line BD.
The center of a circle is eaſily found out by applying the top
of a fquare to the circumference thereof. For if the right-
line GH that joins the points G, H, in which the fides of
the fquare G B, BH, cut the circumference, be bisect-
ed in E, the point E fhall be the center. The demon-
ſtration whereof depends upon Prop. XXXI. of this
Book.
PROP. II. Fig. 2.
If in the circumference of a circle GAC, any two points
A, C, be taken, the right-line AC, which joins those two
points, fall fall within the circle.
Take in the right-line AC any point S; from the cen-
a 15. def. 1. ter G draw GÅ, GS, GC. Becauſe GA (a) =GC,
therefore is the angle A (b) C. But the angle GSC (c)
A, therefore is ĞSC(d)➡C, therefore GC (d)
b 5. 2.
c 16, 2.
d 19. 1.
GS.
But
1
•
EUCLIDE's Elements.
37
But GC only reaches the cirumference, therefore G S
comes not ſo far; wherefore the point S is within the
circle. The fame may be proved of any other point in
the line AC. And therefore the whole line AC falls
within the circle. Which was to be dem.
Coroll.
Hence, if a right-line touch a circle, without cutting
it, it touches but in one point.
PRO P. III. Plate II. Fig. 5.
If in a circle EABC, a right-line BD drawn thro' the
center, bifects any other line AC, not drawn thro' the center,
it fhall alfo cut it at right-angles: And if cuts it at right-
angles, it ſhall alſo biſect the ſame.
From the center E let the lines EA, EC, be drawn.
1. Hyp. Becauſe AF (a) =FC, and EA (EC, and
the fide EF common; the angles EFA, ÈFC, (c) ſhall
be equal, and (d) confequently right. Which was to be
demonftrated.
2. Hyp. Becauſe EFA (e) =EFC, and the angle EAF
(ƒ)=ECF, and the fide EF common; (g) therefore is AF
FC. Therefore AC is cut into two equal parts.
Which was to be demonftrated.
Coroll.
Hence, in any equilateral or Ifofceles triangle, if a
line drawn from the vertical angle biſect the baſe, that
line is perpendicular to it. And on the contrary, a per-.
pendicular drawn from the vertical angle bifects the baſe.
PROP. IV. Fig. 6.
If in a circle ACD, two right-lines AB, CD, cut each
other, and neither of them pass thro' the center E, they ſhall
not cut each other into equal parts.
For if one line paſs thro' the center, 'tis plain it can-
not be bifected by the other; becauſe by hypothefis, the
other does not paſs thro' the center.
If neither of them paſs thro' the center, then from
the center E draw EF; now if AB, CD, were both
bifected in F, then (a) would the angles EFB, EFD, be
both right, and confequently equal. (b) Which is abfurd.
PROP.
C 3
a hyp.
b15.def.1.
c 8. 1.
dio.def.i.
e hyp, and
f
12. ax.
£ 5. 1.
g 26. 1.
a 3. 3.
b 9. ax.
:
38
The third Book of
a 15. def. 1.
b 9. ax.
15. def. 1.
a
b 9. ax.
23. I
8 20. 1.
b 15. def.
c 9. ax.
d 24. 1.
e zo. I.
f 5. ax.
& conf.
4. I.
PROP. V. Plate II. Fig. 7.
If two circles BAC, BDC, cut each other, they ſhall not
have the fame center E.
For otherwiſe the lines EB, EDA, drawn from E, the
common center, would be DE (a) —EB (a) =EA (b)
Which is abfurd.
If
PRO P. VI. Fig. 8.
1
two circles BAC, BDE, inwardly touch each other (in
B) they have not one and the fame center F.
For otherwife the right-lines FB, FDA, drawn from
drawn
the center F, would be FD (a)=FB (a)=FA. (b) Which
is abfurd.
PRO P. VII. Fig. 9.
If in AB, the diameter of a circle, fome point G be
taken, which is not the center of the circle, and from
that point certain right-lines GC, GD, GE, fall on the
circle, the greateſt line fhall be that (GA) which paſf-
eth thro' the center F; the leaſt, the remainder of the
fame line (GB.) And of all the other lines, the line
GC, neareſt to that which was drawn thro' the center is
always greater than any line farther removed GD; and
there can but two equal lines fall from the fame point
on the circle, viz. one on each fide of the leaſt GB, or
of the greateſt GA.
From the center F draw the right-lines FC, FD, FE;
* make the angle BFH-BFE.
1. GF FC (that is GA) (a)GC. Which was to
be demonftrated.
2. The fide FG is common, and FC (b)=FD, and
the angle GFC (c) GFD; (d) wherefore the bafe GC
GD.
3. FB (FE) (e)GE+GF. Therefore FG, which
is common, being taken away from both, there remains
BGEG.
4. The fide FG is common, and FE FH, and the
angle BFH (g) =BFE; (b) Therefore is GE-GH.
But that no other line GD from the point G, can be
equal to GE, or GH, is already proved. Which was to
be demonftrated.
PROP.
EUCLIDE's Elements.
39
;
PROP. VIII. Plate II. Fig. 10.
If fome point A be taken without a circle, and from
that point be drawn certain right-lines AI, AH, AG,
AF, to the circle, and of thoſe one AI, be drawn thro'
the center K, and the others any wife; of all thoſe lines
that fall on the concave of the circumference, that is
the greateſt (AI) which is drawn thro' the center; and
of the others, that (AH) which is neareſt to the line
that paffes thro' the center, is greater than that which
is more diſtant, AG. But of all thofe lines that fall on
the convex part of the circle, the leaft is that (AB)
which is drawn from the point A, to the diameter IB;
and of the others, that (AC) which is neareſt to the
leaſt, is less than that which is farther diftant AD. And
from that point there can be only two equal right-lines
AC, AL, drawn, which fhall fall on the circumference
on each fide of the leaſt line AB, or of the greateſt AI.
From the center K, draw the right-lines KH, KG,
KF, KC, KD, KE, and make the angle AKL:
AKC.
1. AI (AK+KH)(a) — AH.
2. The fide AK is common, and KH KG, and the
angle AKH AKG; (b) therefore the baſe AHC
AG.
a 20. 1.
b 24. I.
C 20. I.
3. KA() KC+CA. From hence take away
~ 7
KČ, KB, which are equal; then will remain AB, (a) d 5. ax.
AC.
4. AC+CK (e)~AD+DK. Take from both, CK,
DK, which are equal; then remains AC (ƒ)~AD.
5. The fide KA is common, and KL KC, and the
angle AKL(g)=AKC; (b) therefore LA CA. But that
no other line could be drawn equal to thefe, was proved
above. Therefore, &c.
PRO P. IX. Fig. 11.
If in a circle BCK, a point A be taken, and from that
point more than two equal right-lines AB, AC, AK, can be
drawn to the circumference, then is that point A the center
of the circle.
For (a) from no point without the center can more
than two equal right-lines be drawn to the circumfe-
rence. Therefore A is the center. Which was to be
demonftrated.
C 4
PROP.
€ 21. I.
f 5. ax.
g confir.
h 4. I.
a 7.3.
40
The third Book of
a cor. 1. 3.
PROP. X. Plate II. Fig. 12.
A circle IAKBL, cannot cut another circle IEKFL, in
more than two points.
Let one circle, if it may be, cut the other in three
points I, K, L, and IK, KL, being join'd, let them be
bifected in M and N. (a) Both circles have their centers
in their perpendiculars MC, NH, and in the interfection.
of thoſe perpendiculars which is O. Therefore the cir-
cles that cut each other have the fame center. Which
is falſe, by Prop. 5. 3.
PROP. XI. Fig. 8.
If two circles FBDE, GABC, touch one the other in-
wardly, and their centers be taken G, F; a right-line FG
joining their centers, and produced, shall cut the circumference
in B, the point of contact of the circles,
If it can be, let the right-line FG produced cut the
circles in ſome other point than B; fo that not GFB,
but GFIK, fhall be a right-line. Let the line FB be
a 15. def. 1. drawn. Now, becauſe FB (a)=FI, and FK (6)GB
b 7.3.
(fince the right-lines GFB paffes thro' G, the center of
the greater circle) therefore is FI - FK. (c) Which is
abfurd.
c g. ax.
2. 20. I.
bg.az.
a 11.3.
b 15. def. 1.
€ 15. def. 1.
d 9. ax.
€ 2. 3›
PROP. XII. Fig. 13.
If two circles ACD, BCE, touch one the other outwardly,
the right-line AB, which joins their centers A, B, fhall pass
thro' the point of contact C.
If it may be, let ADEB be a right-line cutting the
circles, not in the point of contact C, but in the points
D, E; draw AC, CB, then is AD-J-EB (AC+CB) (a)
ADEB. (b) Which is abfurd.
PRO P. XIII. Fig. 8.
A circle GCB cannot touch a circle FBE in more points
than one B, whether it be inwardly or outwardly.
1. Let one circle (if it can be) touch another in two
points B, E, (a) Then will the right-lines GF, that joins
the centers, if it be produced, fall as well in B as E.
Now becauſe GE (b)= GB, and FEGE, therefore
is FB ((c) FE) GB. (d) Which is abfurd.
2. If it be faid to touch outwardly in the points L
and M, then draw the line LM, (e) which will be in both
circles.
EUCLIDE'S Elements.
41
circles. Therefore thofe circles cut one the other;
which is against the Hyp.
PRO P. XIV. Plate II. Fig. 14.
But
In a circle EABC, equal right-lines AC, BD, are equally
diflant from the center E: And right-lines AC, BD, which
are equally diftant from the center, are equal among themſelves.
From the center E, draw the perpendiculars EF, EG,
(a) which will bifect the lines AC, BD; join EA, EB.
1. Hyp. AC BD, therefore AF (b): BG.
alfo EA EB; therefore FEq (c) EAq - AFq=
EBq BGq (c) = EGq. (d) Therefore FE = EĠ.
2. Hyp. EF EG. Therefore AFq (e) EAq-
EFqEBqEGq BGq. Therefore AF (d) =
GB, and (e) conſequently ACBD. Which was to be
demonftrated.
=
PROP. XV. Fig. 15.
In a circle GABC, the greatest line is the diameter AD;
and of all other lines, that FE, which is nearest to the
center G, is greater than any line BC farther diftant from it.
1. Draw GB, and GC. The diameter AD (a) GB
+GC (b) — BC.
2. Let the diſtance GI be GH. Take GN
GH. Thro' the point N draw KL, perpendicular to
GI: Join GK, GL. Becauſe GK = GB, and GL
GC, and the angle KGL BGC; (c) therefore is KL
(FE) BC. Which was to be demonfirated.
PRO P. XVI. Fig. 16.
A line CD, drawn from the extreme point of the diameter
HA, of a circle BALÍ, perpendicular to the faid diameter,
Ball fall without the circle; and between the fame right-
line and the circumference, cannot be drawn another line
AL. And the angle of the femicircle BAI, is greater
than any right-lined acute angle BAL; and the remain-
ing angle without the circumference DAI, is less than
any right-lined angle.
1. From the center B, to any point F, in the right-
line AC, draw the right-line BF. The fide BF fubtend-
ing the right-angle BAF, is (a) greater than the fide BA,
which is oppofite to the acute-angle BFA. Therefore,
whereas BA, (BG) reaches to the circumference, BF,
fhall reach further; and fo the point F, and for the
fame
a 3. 3.
b 7. ax.
C 47. I.
and
3.
d fchol.
ax.
48. 1.
e 6. axi
a 15.def. 1.
b 20. Is
C 24. I.
a 19. f
42
The third Book of
b 19. 1.
a 15. def. 1.
b 4. 1.
c cor. 16 3.
a def. 2. 3.
b cor. 17. 1.
€ 19. 1.
á 9. ax.
fame reafon any other point of the line AC, fhall be
without the circle.
2. Draw BE perpendicular to AL. The fide BA, op-
pofite to the right-angle BEA, is (b) greater than the fide
BE, which fubtends the acute-angle BAE; therefore
the point E, and fo the whole line EA, falls within
the circle.
3. Hence it follows, that any acute-angle, to wit,
EAD, is greater than the angle of contact DAI, and
that any acute-angle BAL is lefs than the angle of a
femicircle BAI. Which was to be dem.
Coroll.
Hence, a right-line drawn from the extremity of the
diameter of a circle, and at right-angles, is a tangent
to the faid circle.
From this propofition are gathered many paradoxes,
and wonderful confectaries, which you may meet with
in the interpreters.
PRO P. XVII. Plate II. Fig. 17.
From a point given A, to draw a right-line AC, which
hall touch a circle given D B C.
From D, the center of the circle given, to the gi-
ven point A, let the line DA be drawn, cutting the cir-
cumference in B, from the center D, defcribe another
circle thro' the point A; and from B, draw a per-
endicular to AD, which ſhall meet with the circle
AE in the point E; and draw ED meeting with the
circle BC, in the point C. Then a line drawn from
A to C, fhall touch the circle DBC.
For DB (a) DC, and DE (a) DA, and the angle
Dis common; (b) therefore the angle ACD EBD
and both right. (c) Therefore AC, touches the circle
in C. Which was to be done,
PRO P. XVIII. Fig. 16.
If any right-line CA touches a circle BALH, and from the
center to the point of contact A, a right-line BA be drawn
that line BA fhall be perpendicular to the tangent CA.
;
If you deny it, let fome other line BF be drawn from
the center B, perpendicular to the tangent, and (a)
cutting the circle in G.
cutting the circle in G. Therefore, whereas the angle
BFA is faid to be right, (b) thence the angle BAF
acute; (c) fo that BA (BG)
BF. (d) Which is abfurd.
PROP.
EUCLIDE's Elements.
43
PROP. XIX. Plate II. Fig. 16.
If any right-line CD touch a circle, and from the point of
contact A, a right-line AH, be erected at right-angles to
the tangent, the center of the circle fhall be in the line AH
fo erected.
If you deny it, let the center be without the line AK,
in the point K; and from K, to the point of contact,
let KA be drawn. Therefore the angle KAD is right,
and (a) confequently equal to the angle HAD, which
was right by Hypothefis. (b) Which is abfurd,
PROP. XX. Fig. 18, 19, 20.
In a circle DABC, the angle BDC at the center, is dou-
ble of the angle BAC at the circumference, when the fame arch
of the circle BC, is the base of the angle.
Draw the Diameter ADE, The outward angle BDE
(a)=DAB+DBA (6) —2DAB : In like manner the an-
gle EDC DAC, Therefore in the first cafe (c) the
whole angle BDC=2BAC, and in the third cafe the
remaining angle BDC (d)=2BAC. Which was to be de-
monftrated.
PROP. XXI, Fig, 21,
In a circle EDAC, the angles DAC, and DBC, which
are in the fame fegment, are equal one to the other.
1. Cafe. If the fegment DABC be greater than a
femicircle, from the center E draw ED, EC. Then is
twice the angle A (a) E (a) = 2 B. Which was to be de-
monftrated.
2. Cafe. If the fegment, (Fig, 22.) be lefs than a femicir-
cle, then is the fum of the angles of the triangle ADF
equal to the ſum of the angles of the triangle BCF; from
each let AFD equal to BFC, (b) and ADB (c) = ACB,
be taken away, then remains DAC-DBC, Which was
to be demonftrated,
PRO P. XXII. Fig. 23.
The angles ADC, ABC, of a quadrilateral figure ABCD,
defcribed in a circle, which are oppofite one to the other, are
equal to two right-angles.
Draw AC, BD. The angle ABC+BCA+BAC (a)
2 right-angles. But BDA (6)=BCA, and BDC (b)
BAC. (c) Therefore ABC+ADC-2 right-angles.
Which was to be demonftrated.
Coroll
a 12. ax.
b9
9. ax.
a 32. I.
b 5. 1.
c 2 ax.
d 20. ax.
a 20. 3.
b 15. 1.
c by the ift
cafe.
a 32. I.
b21.3.
ci. ax.
44
The third Book of
a 22. 3.
b byp.
€ 3. ax.
d 21. 1.
I.
Corol.
Hence, if one fide AB of a quadrilateral, defcri-
bed in a circle, be produced, the external angle EBC
is equal to the internal angle ADC, which is oppofite
to that ABC, and adjacent to EBC, as appears by 13. 1.
and 3. ax.
2. A circle cannot be deſcribed about a Rhombus
becauſe its oppofite angles are greater, or leſs than two
right-angles.
Schol. Plate II. Fig. 24.
If in a quadrilateral ABCD, the angles A, and C, which
are oppofite, be equal to two right-angles, then a circle may
be defcribed about that quadrilateral.
•
For a circle will pass through any three angles B,C,D,
(as appears by 5. 4.) I fay that it fhall alfo pafs thro'
A, the 4th angle of fuch a quadrilateral: For if you
deny it, let the circle paſs thro' F: Therefore the right-
lines BF, FD, BD being drawn, the angle CF (a) =
2 right (6)=C+A; wherefore A (c) is equal to F, (d)
Which is abfurd.
PROP. XXIII. Fig. 25.
Two like and unequal fegments of circles ABC, ADC, can-
not be fet on the fame right-line AC, and on the fame fide
thereof.
For if they are faid to be like, draw the line CB
cutting the circumference in D and B, join AB and AD.
a 10. def. 3. Becauſe the fegments are fuppofed like, (a) therefore is
the angle ADC=ABC. (b) Which is abfurd.
b 16. 1.
a 23.3.
b 10. 3.
c 8. ax.
PRO P. XXIV. Fig. 26.
Like fegments, of circles ABC, DEF, upon equal right-lines
AC, DF, are equal one to the other.The baſe AČ being
laid on the baſe DF, will agree with it, becauſe AC=
DF. Therefore the fegment ABC fhall agree with the
fegment DEF (for otherwiſe it ſhall fall either within or
without) and if fo (a) then the fegments are not like,
which is contrary to the Hypothefis, and at leaſt it ſhall
fall partly within and partly without, and fo cut in three
points, (b) which is abfurd. (e) Therefore the ſegment
ABCDEF. Which was to be demonftrated.
PROP.
EUCLIDE's Elements.
45
PRO P. XXV. Plate II. Fig. 27.
A fegment of a circle AB C, being given, to defcribe
the whole circle whereof that is a fegment.
Let two right-lines be drawn AB, BC, which bi-
fect in the points D and E. From D and E draw
the perpendiculars DF, EF, meeting in the point F.
I ſay this point fhall be the center of the circle.
For the center fhall be as well in (a) DF as EF,
therefore it muſt be in the point F, which is common
to them both. Which was to be done.
PRO P. XXVI. Fig. 28, 29.
In equal circles GABC, HDEF, equal angles ftand up-
on equal parts of the circumference, AC, DF; whether thofe
angles be made at the centers G, H, or at the circumferences,
B, E.
Becauſe the circles are equal, therefore is GA=HD,
and GC=HF; alfo by Hypothefis the angle G=H,
(a) therefore AC-DF. Moreover the angle B (b)=1G
(‹) = { H (b) =E. (d) Therefore the fegments ABC,
DEF are like, and (e) confequently equal; (f) whence
the remaining fegments alfo AC, DF, are equal. Which
was to be demonſtrated.
Schol. Fig. 30.
In a circle ABCD, let an arch AB be equal to DC;
then ſhall AD be parallel to BC. For the right-line AC
being drawn, the angle ACB (a) CAD; wherefore
by 27. 1. the faid fides are parallel.
PROP. XXVII. Fig. 28, 29.
In equal circles GABC, HDEF, the angles ftanding upon
equal parts of the circumference. AC, DF, are equal between
themſelves, whether they be made at the centers Ĝ,H, or at
the circumferences, B, E.
For if it be poffible, let one of the angles AGC be
DHF, and make AGI-DHF; thence is the arch
AI (a) =DF (b) ≈AC. (c) Which is abſurd,
Schol. Fig. 31.
A right-line EF, which being drawn from A the mid-
dle point of any periphery BC, touching the circle, is parallel
to
a cor. 1. 3.
a 4. 1.
b 20. 3.
c hyp.
dio.def.3.
e 24. 3.
f 3. ex.
a 26.3.
a 26. 3.
b byp. I
cg.ax.
46
The third Book of
a 27. 3.
b byp.
to the right-line BC, fubtending the ſaid periphery,
From the center D draw a right-line DA to the point
of contact A, and join DB, DC.
The fide GD is common, and DB-DC, and the
angle BDA (a)=CDA, (becauſe the arches BA, CA are
(b) equal) therefore the angles at the baſe DGB, DGC
are (c) equal, and (d) confequently right; but the inward
d 10. def. 1. angles GAE, GAF are alſo (e) right; (ƒ) therefore BC,
e hyp.
EF are parallel. Which was to be demonftrated.
C 4. I.
f 28. I.
a hyp..
b 8. 1.
c 26. 3.
d. 3. ax.
a hyp.
b. 27. 3.
C4.1.
a confir.
b 12, ax.
€ 4. I.
d 28. 3-
PRO P. XXVIII. Plate IL Fig. 28, 29.
In equal circles GABC, HDEF, equal right-lines AC,
DF, cut off equal parts of the circumference, the greatest
ABC, equal to the greatest DEF, and the leaft ALC to the
leaft DKF.
From the centers G, H, draw GA, GC, and HD, HF.
Becauſe GA-HD, and GC-HF, and AC (a)-DF,
(b) therefore is the angle G=H; (c) whence the arch
ALC=DKF; (d) and fo the remaining arch A B C—
DEF. Which was to be demonftrated.
But if the fubtended line AC be or
DF, then in like manner will be the arch AC be
than DF.
PRO P. XXIX. Fig. 28, 29.
than
or
In equal circles GABC, HDEF, the right-lines AC,
DF, which ſubtend equal peripheries ABC, DEF, are equal.
Draw the lines GA, GC, and HD, HF. Becaufe
GA HD, and GC HF, and (becauſe the arches
AC, DF are (a) equal) the angle G (b) H, (c) there-
fore is the bafe AC=DF. Which was to be demonftrated.
This and the three precedent propofitions may be
underſtood alſo of the fame circle.
PROP. XXX. Fig. 32.
To cut a Periphery given ABC into two equal parts.
Draw the right-line AC, and bifect it in D; from
D draw a perpendicular DB meeting with the arch
in B, it fhall bifect the fame:
=
=
For join AB, and CB. The fide DR is common,
and AD (a) DC, and the angle ADB (b) CDB.
(c) Therefore AB = BC; (a) whence the arch AB=BC.
Which was to be done.
+
PROP.
EUCLIDE's Elements,
47
PROP. XXXI. Plate II. Fig. 33.
In a circle the angle ABC, which is in the femicircle,
is a right-angle; but the angle, which is in the greater
Segment BAC, is less than a right-angle, and the angle
which is in the leffer fegment BFC is greater than a right-
angle. Moreover, the angle of the greater fegment is greater
than a right-angle, and the angle of the leffer fegment is
less than a right-angle.
A+
From the center D draw DB. Becauſe DBDA,
therefore is the angle A (a)= DBA, and the angle DCB
(a) = DBC, (b) therefore the angle A B C
ACB, (c) EBC, (d) fo that ABC and EBC are right-
angles. W. W. D. (e) Therefore BAC is an acute-an-
gle. W.W. D. And further, whereas BAC+ BFC(f)
2 right-angles, therefore BFC is an obtufe-angle.
Laftly, the angle contained under the right-line CB, and
the arch BAC is greater than the right-angle ABC;
but the angle made by the right-line CB, and the pe-
riphery of the leffer fegment BFC (g) is lefs than the
right-angle EBC. Which was to be demonftrated.
Schol.
In a right-angled triangle A B C, if the hypothenufe (or
line fubtending the right-angle) AC be bifected in D, a cir-
cle drawn from the center D, through the point A, ſhall
alfo pass through the point B; as you may eaſily demon-
Atrate from this prop. and 21. I.
PRO P. XXXII. Fig. 34.
If a right-line AB touch a circle, and from the point of
contact be drawn a right-line CE, cutting the circle,
the angles ECB, ECA, which it makes with the tangent
line, are equal to those angles EDC, EFC, which are
made in the alternate fegments of the circle.
(ƒ)
Let CD, the fide of the angle EDC be perpendi-
cular to AB, (a) for it is to the fame purpoſe, (b) there-
fore CD is the diameter, (c) therefore the angle CED in a
femicircle is a right-angle, (d. and therefore the angle
DDCE to a right-angle (e) ECB+ DCE. (f)
Therefore the angle DECB, Which was to be dem.
Now whereas the angle ECB+ ECA (g)= 2 right-
angles (b) = DF, from both of thefe take away
ECB and D), which are equal, (4) there remains ECA
F. Which was to be dem.
PROP.
a 5. 1.
b2.ax.
C 32. I.
dio.def.1.
e cor. 17. I.
f 22. 3.
g 9. ax.
a 26. 3.
b 19. 3-
C 31. 3.
d 3z. 1.
e conſtr.
£ 3. ax.
g 13. 1.
h 22. 3.
k 3. ax.
48
The third Book of
a 23. 1.
c conftr.
c 6. I.
cor. 16.
d cor.
a 32. 3.
£ conſtr.
a 17.3.
b 23. 1.
c 32. 3.
d conftr.
a 5. 2.
3
PROP. XXXIII. Plate II. Fig. 35.
Upon a right-line AB to defcribe a fegment of a circle
AIEB which shall contain an angle AIB, equal to a right-
lined angle given C.
(a) Make the angle BAD C. Through the point A
draw the line AE perpendicular to HD. At the other
end of the line given AB make an angle ABF BAF,
one of the fides of which fhall cut the line AE in F;
from the center F, through the point A, defcribe a
circle, which fhall pass through B. (Becauſe the angle
FBA (6) =FAB, and (c) therefore FB-FA) AIB is
the feginent fought. For becauſe HD is perpendicular
to the diameter AE, therefore HD (d) touches the circle
which AB cuts, And therefore the angle AIB (e) =
BAD (ƒ)=C. Which was to be done.
PRO P. XXXIV. Fig. 36.
From a circle given ABC to cut off a fegment ABC con-
taining an angle В, equal to a right-lined angle given D.
(a) Draw a right-line EF which fhali touch the circle
given in A, (b) let AC be drawn making an angle FAC
D. Alfo draw (6) AB, making the angle BAED,
and join B, C. The line AB fhall cut off the ſegment
ABC containing an angle B (c) CAF (d)=D. Which
was to be done.
PROP. XXXV. Fig. 37.
If in a circle DBCA two right-lines AB, DC cut each
other, the rectangle comprehended under the fegments AE, EB,
of the one, ſhall be equal to the rectangle comprehended under
the fegments CE, ED of the other.
1. Cafe. If the right-lines cut one the other in the cen-
ter, the thing is evident.
2. Cafe. If one lineAB (Fig. 38.) paffes thro' the center
F, and bifects the other line CD, then draw FD. Now
bſch. 48. 1. the rectangle AEB+FEq (a) =FBq. (b) =FDq. (c):
EDq+FEq (a) =CEDFEq. (e) Therefore the rectan-
gle AEB⇒CED. Which was to be demonftrated.
€ 47. I.
d hyp.
€ 3. ax.
=
3. Cafe If one of the lines AB (Fig. 39.) be the dia-
meter, and cut the other line CD unequally, bifect CD
by FG, a perpendicular from the center.
Theſe
EUCLIDE's Elements.
49
The rectangle AEB+FEq.
Thefe
(ƒ) FBq (FDq)
are
(g) FGq + GDq
equal.
FGq+(6) GEq+Rectang. CED.
(k) FEq+CED.
(4) Therefore the rectangle AEB-CED.
4. Cafe. If neither of the right-lines AB, CD pafs thro'
the center, then through the point of interfection E, draw
the diameter GH. By that which hath been already
demonftrated it appears, that the rectangle AEB=GEH
=CED. Which was to be demonftrated.
More eaſily, and generally, thus; join AC and BD,
then becauſe the angles (a) CEA, DEB, and (b) alfo C, B
(upon the fame arch AD) are equal, thence are the tri-
angles CEA, BED, (c) equiangular. (d) Wherefore CE :
EA EB: ED, and (e) confequently CEX ED-AEX
EB. Which was to be demonftrated.
The citations out of the 6 Book, both here and in
the following prop, have no dependance on the fame; ſo
that it was free to uſe them.
PROP. XXXVI. Plate II. Fig. 42.
If any point be taken without a circle EBC, and from
that point two right-lines DA, DB, fall upon the circle,
whereof one DA cuts the circle, the other DB touches it,
the rectangle comprehended under the whole line DA that
cuts the circle, and DC, that part which is taken from the
point given D to the convex of the periphery, fhall be equal
to the fquare made of the tangent line.
1. Cafe. If the fecant AD paffes thro' the center, then
join EB, this (a) will make a right-angle with the line
DB, wherefore DBq+ (d) EBq (ECq,) (b) =EDq (c) =
ADX DCECq. Therefore AD XDC=DBq. Which
was to be demonstrated.
f
£ 5.2.
8 47. 1.
h
5.2.
k
47. I.
1
3. ax.
Fig. 40.
Fig. 41,
a 15. 1.
b 21.3.
ccor. 32. 1.
d 4. 6.
e 16.6.
a 18. 3.
b 47. 1.
c 6. 2.
d 3. ax.
2. Cafe. But if AD paffes not thro' the center, then Fig. 43.
draw EC, EB, ED, and EF perpendicular to AD, (a)
wherefore AC is bifected in F.
=
Becaufe BDq+EBq (6) ≈DEq (6) EFq+FDq (c)
EFq-t-ADC+FCq (d)=ADC+CEq (EBq.) (e) There-
fore is BDq ADC. Which was to be demonftrated.
More eafily, and generally thus; draw AB and BC.
Then, becauſe the angles A, and DBC (a) are equal,
and the angle D common to both; thence are the trian-
gles BDC, ADB (b) equiangular. (c) Wherefore AD:
DB DB: CD; and (d) confequently ADXDC=DBq:
Which was to be demonftrated.
D
Coroll.
a 3. 3.
b 47. I.
c 6. z.
d 47. I.
e 3. ax.
Fig. 44.
a 32. 3.
b 32. 1.
€ 4.6.
d 17. 6.
1
50
The third Book of &c.
a 36. 3.
a 36. 3.
b 36. 3.
C 2. cor.
d 8. 3.
e 2. cor.
f hyp.
g 8.3.
a 17. 3.
b hyp.
c 36. 3.
dx. ax. &
ſch. 4. 1.
e 8. I.
f 12. ax.
Coroll. Plate II.
Fig. 45:
1. Hence, If from any point A, taken without a circle,
there be ſeveral lines AB, AC drawn which cut the circle;
the rectangle comprehended under the whole lines AB,
AC, and the outward parts AL, AF, are equal between
themſelves.
g cor. 16. 3.
h 8. 1.
For if the tangent AD be drawn, then is CAF-ADq
(a) =BAE.
2. It appears alfo from hence, that if two lines AB,
AC, (Fig. 46.) drawn from the fame point do touch a
circle, thofe two lines are equal one to the other.
For if AE be drawn cutting the circle, then is ABq
(a) = EAF (b) =ACq,
3. It is alfo evident, that from a point A, taken with-
out a circle, there can be drawn but two lines AB, AC,
that fhall touch the circle.
For if a third line AD be faid to touch the circle,
thence is AD ()=AB (c)=AC. (d) Which is abfurd.
4. And, on the contrary, it is plain, that if two equal
right-lines A B, A C, fall from any point A, upon the
convex periphery of a circle, and that if one of theſe
equal lines A B touch the circle, then the other A C
touches the circle alfo.
For if poffible, let not A C, but another line A D,
touch the circle; therefore is AD (e) =AC (ƒ) =AB.
(g) Which is abfurd.
PROP. XXXVII. Fig. 47.
If without a circle EBF any point D be taken, and from
that point two right-lines DA, DB fall on the circle, whereof
one line DA cuts the circle, the other DB falls upon it; and
if alfo the rectangle comprehended under the whole line that
cuts the circle, and under that part of it DC which is taken
betwixt the point D and the convex periphery, be equal to
that fquare which is made of the line DB falling on the circle,
I fay that that line DB fo falling fhall touch the circle given.
From the point D (a) let a tangent DF be drawn, and
from the center E draw ED, EB, EF. Now becauſe
DBq (6) ADC (c)=DFq, therefore is DB (d)=DF:
But EB-EF, and the fide ED common; (e) therefore
the angle EBD EFD; but EFD is a right-angle, and
(ƒ) therefore EBD is right alſo; and (g) therefore DB
touches the circle. Which was to be dem.
Coroll.
From hence it follows that the (b) angle EDB EDF,
The End of the third Boo K.
}
Į
[ 51 ]
The FOURTH BOOK
O F
EUCLIDE's
I.
ELEMENT S
"Α
Definitions.
Right-lined figure is faid to be infcribed in
a right-lined figure, when every one of the
angles of the infcribed figure touch every
one of the fides of the figure wherein it is inſcribed.
So the triangle DEF is infcribed in the triangle ABC.
Plate II. Fig. 48.
II. In like manner a figure is faid to be deſcribed
about a figure, when every one of the fides of the figure
circumfcribed, touch every one of the angles of the figure
about which it is circumfcribed.
So the triangle ABC is defcribed about the triangle DEF.
III. A right-lined figure is faid to be infcribed in a
circle, when all the angles of that figure which is in-
ſcribed touch the circumference of the circle. As Fig. 50.
IV. A right-lined figure is faid to be deſcribed about a
circle, when all the fides of the figure which is circum-
fcribed touch the periphery of the circle. As Fig. 49.
V. After the like manner a circle is faid to be infcri-
bed in a right-lined figure, when the periphery of the
circle touches all the fides of the figure, in which it is
infcribed. Fig. 49.
VI. A circle is faid to be deſcribed about a figure
when the periphery of the circle touches all the angles
of the figure, which it circumfcribes.
VII. A right-line is faid to be fitted or applied in a
circle when the extremes thereof fall upon the circumfe-
rence; as the right-line AB. Fig. 6.
PROP. I. Fig. 51.
In a circle given ABC to apply a right-line AB equal to a
D 2
rights
52
The fourth Book of
a 3. post. &
3. I.
b 15. def. 1.
c conftr.
a 17. 3.
b23. I.
c 32. 3.
d conſtr.
e 32. I.
a 23. 1.
b 17. 3.
c 13. ax
dII.
d 11. 3.
e ſch, 32. 1.
f 13. ax.
g conſtr.
h 3. ax.
k 32. I
2 9. I
biz.
b 12.
right-line given D, which doth not exceed AC, the diameter
of the circle.
From the center A at the the diſtance AED (a) de-
ſcribe a circle meeting with the circle given in B, draw
Then is AB (6) AE (c)—D. Which was to be
AB.
done.
PROP. II. Plate II. Fig. 49, 50.
In a circle given ABC to defcribe a triangle ABC, equian-
gular to a triangle given DEF.
Let the right-line GH (a) touch the circle given in A;
(6) make the angle HACE, (6) and the angle GAB=F,
then join BC; and the thing is done.
For the angle B (c)=HAC (d)=E, and the angle C
(c) =GAB (d) =F; (e) whence alfo the angle BACD.
Therefore the triangle B A C inſcribed in the circle is
equiangular to DEF. Which was to be done.
PROP. III. Fig. 52, 53.
About a circle given IABC to deſcribe a triangle LNM,
equiangular to a triangle given DEF.
Produce the fide EF on both fides; at the center I (a)
make an angle AIB-DEG, and an angle BIC=DFH.
Then through the points A, B, C, let three right-lines
LN, LM, NM, (b) be drawn, touching the circle, and the
thing is done.
For it's evident that the right-lines LN, LM, MN,
will meet and make a triangle, (c) becauſe the angles
LAI, LBI are right; fo that if the (d) right-line AB
was drawn, it would make the angles LAB, LBA, leſs
than two right-angles.
Since therefore the angle AIB-+L (e)=2 right-angles
(f)=DEG+DEF, and AIB (g)=DEG; (b) therefore
is the angle L-DEF. By the fame way of reaſoning
the angle MD FE, (k) Therefore alfo the angle N=
D. And therefore the triangle LNM, deſcribed about
the circle, is equiangular to EDF, the triangle given.
Which was to be done.
PRO P. IV. Fig. 54.
In a triangle given ABC, to defcribe a circle EFG.
(a) Biſect the angles B and C with the right-lines BD,
CD, meeting in the point D, (b) and draw the perpen-
diculars DE, DF, DG. A circle defcribed from the
center
EUCLIDE's Elements.
53
center D through E, will pafs through G and F, and
touch the three fides of the triangle.
For the angle DBE (c) =DBF; and the angle DEB (d)
=DFB; and the fide DB common, (e) therefore DE=
DF. For the fame reafon DG-DF, The circle there-
fore defcribed from the center D paffes through the
three points E, F, G; and whereas the angles at E, F,
G, are right, therefore it touches all the fides of the
triangle. Which was to be done.
Schol.
c conftr.
diz ax,
e 26. 1.
Hence, The fides of a triangles being known, their feg. Pet. Herig.
ments which are made by the touching of the circle inſcribed,
fhall be found, Thus ;
Let AB be 12, AC 18, BC 16, then is AB+BC=28.
Out of which fubduct 18 AC
AE+FC, there re-
mains 10 BE+BF. Therefore BE, or BF = 5; and
confequently FC, or C G11. Wherefore GA, or
AE, =
= 7.
PROP. V. Plate II. Fig. 55, 56.
About a triangle given ABC, to deſcribe a circle FABC.
(a) Bisect any two fides BA, CA, with perpendiculars
DF, EF, meeting in the point F. I fay this fhall be
the center of the circle,
Now
For, let the right-lines FA, FB, FC be drawn.
becauſe AD (b)=DB, and the fide DF common, and
the angles FDA (c) =FDB, therefore is FB (a) = FA.
After the fame manner is FC FA. Therefore a circle
deſcribed from the center F fhall pass through the an-
gles of the triangle given (viz.) B, A, C. Which was
to be done.
Coroll.
* Hence, if a triangle be acute-angled, the center «
fhall fall within the triangle; if right-angled, in the
fide oppofite to the right-angle, and if obtufe-angled,
without the triangle.
Schol.
By the fame method may a circle be defcribed, that
fhall pass through three points given, not being in the
fame ftrait-line,
a 1o. &
II. I.
b conftr.
c confir. &
12. ax.
d 4. 1.
31. 3.
D 3
PROP.
簪
​54
The fourth Book of
1
a II. I,
b 26.3.
€ 29.3.
d 31. 3.
e29. def. 1.
a 17.3.
b'18. 3.
c 28. I.
d 34. I.
e 15. def. 1.
f 10. def. 1.
8 7.ax.
b byp.
33. I.
PROP. VI. Plate II. Fig. 57.
In a circle given EABCD to infcribe a fquare ABCD.
(a) Draw the diameters AC, BD cutting each other at
right-angles in the center E. Join the extremes of theſe
diameters with the right-lines AB, BC, CD, DA. And
the thing is done.
Now becauſe the four angles at E are right, the (b)
arches and (c) fubtended lines AB, BC, CD, DA, are
equal; therefore is the figure ABCD equilateral, and
all the angles in femicircles, and fo (d) right. (e) There-
fore ABCD is a fquare infcribed in a circle given.
Which was to be done.
PROP. VII. Fig. 58.
About a circle given EABCD, to defcribe a ſquare FHIG.
Draw the Diameters AC, BD, cutting one the other
at right-angles; through the extremes of thefe diameters
(a) draw tangents meeting in F, H, I, G. And the thing
is done.
For becauſe (b) the angles A and C are right, (c) there-
fore is FG parallel to HI. After the fame manner is
FH parallel to GI, and therefore FHIG is a Pgr.
and alſo right-angled. It is equilateral becauſe FG (d)
=HI (d)=DB (e)=CA (d)=FH (d)=GI.
Wherefore FHIG is a (ƒ) fquare circumfcribed to the
circle given. Which was to be done.
Schol. Fig. 59.
A fquare ABCD deſcribed about a circle is double of
the fquare EFGH infcribed in the fame circle.
For the rectangle HB-2 HEF and HD2 HGF, by
the
41. I.
PRO P. VIII. Fig. 58.
In a fquare given FGHI, to infcribe a circle EABCD.
Bifect the fides of the fquare in the points B,D,A,C,
cutting one the other in E, a circle drawn from the cen-
ter E, thro' A, fhall be inſcribed in the fquare.
For becauſe FA and HC are (a) equal and (b) parallel,
(c) therefore is FH parallel to AC, parallel to GI. Af-
ter the fame manner is FG parallel to BD, parallel to
HI; therefore EF, EG, EH, EI, are parallelograms.
There-
EUCLIDE's Elements,
55
Therefore FA (d) =FB (e)=AE=BE=CE=ED. The
circle therefore deſcribed from the center E, through A
ſhall pass through A, B, C, D, and touch the fides of
the fquare, fince the angles A, B, C, D, are right.
Which was to be doee.
PRO P. IX. Plate II. Fig. 57:
About a fquare given ABCD, to defcribe a circle EABCD.
Draw the diagonals AC, BD, cutting one the other
in E. From the center E through A defcribe a circle;
I ſay this circle is circumfcribed to the fquare.
For the angles ABD and BAC are (a) half of right-
angles; (b) therefore EA=EB. After the fame manner
is EA EDEC. The circle therefore defcribed from
the center E paffes through A, B, C, D the angles of the
fquare given. Which was to be done.
PROP. X. Fig. 60.
To make an Ifoceles triangle ABD, having each angle at
the bafe B, and ADB double to the remaining angle A.
d 7.ax.
e 34. 1.
a 4. cor..
32. I.
b 6. 1.
a 11. 2.
Take any right-line AB, and divide it in C, (a) ſo that
ABXBC may be equal to ACq. From the center A thro'
B, deſcribe the circle ABD; and in this circle (b) apply b 1. 4.
BD=AC, and join AD; I fay ABD is the triangle
required.
For draw DC, and through the points C, D, A, (c)
draw a circle. Now becaufe ABXBC=ACq=BDq, (d)
it is evident that BD touches the circle ACD which
CD cutteth; (e) therefore is the angle BDC=A, and
therefore the angle BDC+CDA (f)=A+CDA (g) =
BCD. But BDC+ CDA = BDA (h)
BDA (h) = CBD; (k)
therefore the angle BCD CBD, and therefore DC (1)
=DB= (m) AC; (2) wherefore the angle CDA = A
BDC, therefore ADB = 2 A= ABD. Which avas
to be done.
Coroll.
Whereas all the angles A, B, D, (5) make up two
right-angles, it's evident that A is one fifth of two
right-angles.
PROP. XI. Fig. 61, 62.
In a circle given ABCDE to infcrihe a Pentagon
ABCDE equilateral and equiangular.
(a) Deſcribe
D' 4
c 5.4.
d 37. 3.
e 32. 3.
f 2. ax.
g 32. I.
5. 1.
h
k 1. ax.
1 6. 1.
m conftr..
n 5. I.
h 32. 1.
56
The fourth Book of
a 10. 4.
b 2.4.
€ 9. I.
d 26. 3.
e 29. 3.
£ 27.3.
8 z. ax.
(a) Defcribe an Ifofceles triangle FGH, having each
angle at the baſe double to the other; to the circle, (6)
infcribe a triangle CA D equiangular to the faid tri-
angle F G H. (c) Bifect the angles at the bafe ACD
and ADC with the right-lines DB, CE meeting with
the circumference in B and E, join the right-lines
CB, BA, AE, ED. Then I say it is done.
For it is evident by conftruction, that the angles.
CAD, CDB, BDA, DCE, ECA, are equal; where-
fore the (d) arches and (e) the lines fubtending them DC,
CB, BA, AE, DE, are equal. Therefore the penta-
gon is equilateral, and equiangular, (f) becauſe the an-
gles of it BAE, AED, &c. ftand on equal (g) arches
BCDE, ABCD, &c.
A more eafy practice of this problem fhall be deliver'd
at 10. 13.
Coroll.
Hence, each angle of an equilateral and equiangular
pentagon is equal to three-fifths of two right-angles,
or fix-fifths of one right-angle.
Schol. Fig. 63.
Generally all figures whofe number of fides is odd, are
Pet. Herig. infcribed in circles by the help of Ifofceles triangles, whofe
angles at the baſe are multiples of thofe at the top and
figures whoſe number of fides is odd, are infcribed in a
circle by the help of Ifofceles triangles, whofe angles at
the baſe are multiples fefquialter of thofe at the top.
a 11. 4.
As in the Ifofceles triangle CAB if the angle A≈¿C
=B, then will AB be the fide of a Heptagon. If A=4C,
then is AB the fide of an Enneagon. But if AC, then
is AB the fide of a fquare. And if A≈2 CAB will fub-
tend the fixth part of the circumference, and likewiſe if
A 3C then will AB be the fide of an Octagon.
PRO P. XII. Plate III. Fig. 1.
About a circle given FABCDE, to defcribe a pentagon
HIKLG, equilateral and equiangular.
(a) Inſcribe a pentagon ABCDE in the circle given;
and from the center draw the right-lines FA, FB, FC,
FD, FE; and to thofe lines draw fo many perpendicu-
lars GAH, HBI, ICK, KDL, LEG, meeting in the
points H, I, K, L, G, and the thing is done. For be-
caufe
EUCLIDE's Elements.
+ 57
caufe GA, GE from the fame point G (b) touch the circle,
(c) therefore is GA=GE, and (d) therefore the angle GFA
EGFE, therefore the angle AFE = 2 GFA. After
the fame manner is the angle AFHHFB, and confe-
quently the angle AFB 2 AFH. (e) But the angle
AFE AFB, (f) therefore the angle GFA AFH.
But alfo the angle FAH (g) FAG, and the fide FA is
common, (b) therefore HA=AG=GE=EL, &c. (k)
Therefore HG, GL, LK, KI, IH, the fides of the penta-
gon are equal, and fo alſo are the angles, becauſe double
of the equal angles AGF, AHF, therefore, &c.
Coroll.
After the fame manner, if any equilateral and equi-
angled figure be deſcribed in a circle, and at the extreme
points of the femi-diameters, drawn from the center to
the angles, be drawn perpendicular lines to the faid dia-
meters; I fay, that theſe perpendiculars fhall make ano-
ther figure of as many equal fides and equal angles, cir-
cumfcribed to the circle.
PROP. XIII. Plate III. Fig. 2.
In an equilateral and equiangular pentagon given ABCDE
to inſcribe a circle FGHK.
(a) Bifect two angles of the pentagon A and B, with
the right-lines AI, BK, meeting in the point F. From
F draw the perpendiculars FG, FH, FI, FK, FL. Then
a circle deſcribed from the center F through G will touch
all the fides of the pentagon.
Draw FC, FD, FE. Becaufe BA (6) BC, and the
(6)=BC,
fide BF common, and the angle FBA(c)=FBC, (d) there-
fore is AF FC, and the angle FAB FCB, but the
angle FAB (e) =½ BAE=1 BCD. Therefore the angle
FCB= BCD. After the fame manner are all the whole
angles C, D, E bifected. Now whereas the angle FGB (ƒ)
FHB, and the angle FBH-FBG, and the fide FB
is common, (g) therefore is FG FH. In like manner
are all the right-lines FH, FI, FK, FL, FG equal.
Therefore a circle defcribed from the center F, through
G, paffes through the points H, I, K, L, and (b) touches
the fide of the pentagon, becauſe the angles at thoſe
points are right. Which was to be done.
Coroll.
Hence, if any two neareſt angles of an equilateral and
equianglar figure are bifected, and from that point in
which
bear.16.3.
c2.cor. 36.
d 8. 1.
e 27.3.
£ 7. ax.
g 12. ax.
ĥ 26.1.
k 2. ax.
a g. 1.
b hyp.
c conftr.
d 4. I.
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f 12. aX-
g 26. 1.
h cor.16.3.
58
The fourth Book of
}
a cor. 13. 4.
b 6, 1.
C
a cor. 32. 1.
b 15. 1.
C cor. 13. 1.
d 26. 3.
e 29. 3.
₤27.3.
which the lines meet that bifect the angles, be drawn.
right-lines to the remaining angles of the figure, all the
angles of the figure fhall be bifected,
Schol.
By the fame method may a circle be inſcribed in any
equilateral and equiangular figure.
PRO P. XIV. Plate III. Fig. 1.
About a pentagon given ABCDE, equilateral and equian-
gular, to defcribe a circle FABCDE.
Biſect any two angles of the pentagon with the right-
lines AF, BF, meeting in the point F; the circle de-
fcribed from the center F through A fhall be deſcribed
about the pentagon.
For let FC, FD, FE be drawn. (a) Then the angles C,
D, E are biſected; (b) and therefore FA, FB, FC, FD,
FE are equal; therefore the circle defcribed from the
center F paffes through A, B, C, D, E, all the angles
of the pentagon. Which was to be done.
Schol.
By the fame method is a circle defcribed about any
figure which is equilateral and equiangular.
PROP. XV. Fig. 3.
In a circle given GABCDEF to infcribe an Hexagon (or
fix-fided figure) ABCDEF equilateral and equiangular.
Draw the diameter AD; from the center D through
the center G deſcribe a cirrle cutting the circle given in
the points C and E. Draw the diameters CF, EB; and
join AB, BC, CD, DE, EF, FA. Then I say it's done.
For the angle CGD (a) of 2 right-angles (a)=DGE
(b)=AGF (b)=AGB. (c) Therefore BGC of 2 right-
angles-FGE; therefore the (d) arches and (e) fubtenfes
AB, BC, CD, DE, EF, are equal. Therefore the hexa-
gon is equilateral; but it is equiangled alſo, (ƒ) becauſe
all the angles of it ftand upon equal arches.
1.
Corol.
Hence, the fide of an hexagon inſcribed in a circle
is equal to the femidiameter,
2. Hereby
Plate II. Facing Pag. 58.
S
R
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Fig 1.
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B
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Fig. 3.
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Fig. 16.
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Fig. 19.
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Fig. 20.
B
Fig. 29.
Fig. 24. Fig. 28.
F
Fig. 23.
A
CIDE
B
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Fig.25
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Fig.26.
B
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Fig. 29.
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40
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Fig.36.
F
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45
Fig. 46.
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Fig. 47.
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Fig 5/0.
Fig.54
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Fig.55.
D
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FF
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Fig.59 H
Fig. 61.
B
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Fig. 62.
Fig.63.
Fig.58.
D G
EUCLIDE's Elements.
59
2. Hereby an equilateral triangle ACE may very eafi-
ly be deícribed in a circle given.
Schol. Probl.
To make a true hexagon upon a right-line given CD.
(a) Make an equilateral triangle CGD upon the line
given CD; from the center G through C and D deſcribe
a circle. That circle fhall contain the hexagon made
upon the given line CD.
PRO P. XVI. Plate III. Fig. 4.
In a circle given AEBC, to infcribe a quindecagon (or fif-
teen-fided figure) equilateral and equiangular.
(a) Infcribe an equilateral pentagon AEFGH in the cir-
cle given, and (b) alſo an equilateral triangle ABC, then
I fay BF is the fide of the quindecagon required.
For the arch AB (c) is or of that periphery where-
of AF is or, therefore the remaining part BF is
Ī
of the periphery; and therefore the quindecagon,
whofe fide is BF, is equilateral; but it is equiangular
alfo (d) becauſe all the angles ftand on equal arches of a
circle, whereof every one of the whole circumfe-
recence. Therefore, &c.
A circle is geo-
metrically di-
vided into
parts
S
5
Schol.
1.
9, I
4, 8, 16, &c. by 6, 4, and
3, 6, 12, &c. by 15, 4, and 9, 1.
5, 10, 20, &c. by 11, 4, and 9, 1.
15, 30, 60, &c. by 16, 4, and 9, 1.
Any other way of dividing the circumference into
any parts given is as yet unknown; wherefore in the
conftruction of ordinate figures, we are forced to have
recourſe to mechanic artifices, concerning which you
may confult the writers of practical Geometry.
Andr.
Tacq.
a 1. 1.
a 11.4.
b 2.
4.
c conſtr.
d 27. 3.
The End of the fourth Book.
The
[ 60 ]
The FIFTH BOOK
O F
EUCLIDE's
ELEMEN T S.
A
Definitions.
Part, is a magnitude of a magnitude, a lefs
of a greater, when the lefs meaſureth the
greater.
II. Multiple, is a greater magnitude in
refpect of a leffer, when the leffer meafureth the greater.
III. Ratio, is the mutual habitude or reſpect of two
magnitudes of the fame kind each to other, according
to quantity.
In every ratio that quantity which is referr'd to another
quantity, is called the antecedent of the ratio, and that to
which the other is referr'd, is called the confequent of the
ratio, as in the ratio of 6 to 4, 6 is the antecedent and 4
the confequent.
Note, The quantity of any ratio is known by dividing the
antecedent by the confequent; as the ratio of 12 to 5 is ex-
I
A
preſſed by 1; or the quantity of the ratio of A to B is -.
B
Wherefore, often for brevity fake we denote the quantities of
A
ratio's thus ;
B
C
or, or
that is, the ratio
D
of A to B is greater, equal, or less than the ratio of C to
D. Which must be well obferved by those who would under-
ftand this Book.
Concerning the divers fpecies of ratio's, you may please
to confult interpreters.
IV Proportion is a fimiltude of ratio's.
That which is here termed proportion, is more properly
called proportionality or analogy for proportion commonly
denotes no more than the ratio betwixt two magnitudes.
V. Thof
EUCLIDE's Elements.
61
V. Thofe numbers are faid to have a ratio betwixt
them, which being multiplied may exceed one the other.
E, 12 A, 4. B, 6.| G, 24.
F, 30 C, 10 D, 15. H, 60.
›
VI. Magnitudes are faid
to be in the fame ratio,
the first A, to the fecond B, and the third C, to the fourth
D, when the equimultiples E and F of the firft A,
and the third C, compared with the equimultiples
G, H, of the fecond B, and the fourth D, according
to any multiplication whatfoever, either both together
E, F are leſs than G, H both together, or equal taken
together, or exceed one the other together, if thofe be
taken E, G, and F, H, which anſwer one to the other.
The note hereof is : : ; as A: B:: C: D. That is, as A
is to B, fo is C to D, which fignifies that A to B, and C to
D, are in the fame ratio. We fometimes thus exprefs it
C
A
B
D
that is, A: B::C:D.
VII. Magnitudes that have the fame ratio (A : B :
C: D, are called proportional.
VIII. When of equimul-
E, 30. A, 6. B, 4.G, 28. tiples, E, the multiple of
F, 60. C, 12. D, 9. H, 63. the first magnitude A, ex-
ceeds G, the multiple of the ſecond B, but F, the mul-
tiple of the third C, exceeds not H, the multiple of
the fourth D, then the firſt A to the ſecond B,´has a
greater ratio than the third C to the fourth D.
A
C
If-,-, it is not neceſſary from this definition, that
D
B
E should always exceed G, when F is less than H; but
it is granted that this may be.
IX. Proportion confifts in three terms at leaſt.
Whereof the fecond fupplies the place of two.
X. When three magnitudes A,B,C, are proportional,
the firſt A is ſaid to have a duplicate ratio to the third
C, of that it hath to the fecond B: But when four mag-
nitudes A, B, C, D are proportional, the firſt A is faid to
have a triplicate ratio to the fourth D, of what it has to
the fecond B; and ſo always in order one more, as the
proportion fhall be extended.
Duplicate ratio is thus expreſſed
A
A
twice, that is,
C
B
the ratio of A to C is duplicate of the ratio of A to B.
Triplicate
62
The fifth Book of
A
A
Triplicate ratio is thus expreffed; -- thrice; that
D B
is, the ratio of A to D is triplicate of the ratio of A to B.
Denotes continued proportionals; as A, B, C, D,
2, 6, 18, 54, are continual proportionals.
alfo
XI. Homologous, or Magnitudes of like ratio, are
antecedents to antecedents, and confequents to confe-
quents; that is, if A:B::C: D; A and C; as well as
B and D are called homologous magnitudes.
XII. Alternate proportion is the comparing of ante-
cedent to antecedent, and confequent to confequent. As
if A: B::C: D. therefore alternately, or by permutation,
As ACB: D. by the 16, of 5.
In this definition, and the five following, names are given
to the fix ways of arguing which are often uſed by Mathema-
ticians: the force of which inferences depends on the propofi-
tions of this Book, which are named in their explications.
XIII. Inverſe ratio is when the conſequent is taken
as the antecedent, and fo compared to the antecedent
as the confequent; as A: B: C: D; therefore inverfly
B:A::D:C. by cor. 4. 5.
XIV. Compounded ratio is when the antecedent and
confequent taken both as one, are compared to the con-
fequent it felf. As A: B:: C: D; therefore by compofition
A+BBC+D: D, by 18. 5.
XV. Divided ratio is when the excefs wherein the
antecedent exceedeth the confequent, is compared to
the confequent. As A:B::C:D; therefore by divifion,
A-B:B::C - D: D, by 17. 5.
XVI. Converfe ratio is when the antecedent is compa-
red to the excess wherein the antecedent exceeds the
confequent. As. A:B::C: D; therefore by converſe ratio
A: A~B::C: C-D, by the coroll. of the 19 of the 5.
XVII. Proportion of equality is where there are ta-
ken more magnitudes than two in one order, and alſo
as many magnitudes in another order, comparing two
to two being in the fame ratio; it follows that as in the
firſt order of magnitudes, the firft is to the laft, fo in the
fecond order of magnitudes, is the firſt to the laſt. Or
otherwiſe it is compariſon of the extremes together,
the mean magnitudes being omitted.
Thus let A, B, C, be three magnitudes and D, E, F, three
others, and taking them two by two, let them be in the
fame proportion, that is, let A: B :: D: E, and B: C:
E: F; now if it be inferr'd that A, the first of the first or-
&
der
EUCLIDE's Elements.
63
der, is to C the laf, as D the first of the fecond order, is
to F the last, this form of arguing is faid to be ex æquo,
or from equality.
XVIII. Ordinate proportion is, when antecedent is
to confequent, as antecedent to confequent, and as the
confequent is to any other, fo is the confequent to any
other. As when A:B: D: E. alſo B:C:
B:C::E:F.
and then it shall be A: C: D: F, by the 22. of the 5.
XIX. Perturbate proportion is, when three magni-
tudes being put, and others alfo, which are equal to
theſe in multitude, as in the firft magnitudes the antece-
dent is to the confequent, fo in the fecond magnitudes
is the antecedent to the confequent: and as in the firſt
magnitudes the confequent is to any other, fo in the
fecond magnitudes is any other, to the antecedent. Thus
if A, B, C, and E, F, G, are trvo fets of magnitudes, if
A the first of the first fet, is to B the fecond, as F the fe-
cond of the fecand fet, is to G the last; and alſo if В the
fecond of the first fet is to С the lafi, as E the first of the fe-
cond fet is to F the fecond, fuch is called pertubate proportion,
and by the 23. 5. A: C: : E: G.
XX. Any number of magnitudes being put; the pro-
portion of the firft to the laft is compounded out of the
proportions of the firft to the fecond, the fecond to the
third, and the third to the fourth, &c. to the laft.
Let there be any number of magnitudes A, B, C, D,
A B C
by this definition
Α
II
X
DBC D.
Axiom.
Equimultiples of the fame, or of equal magnitudes
are equal to each other.
PROP. I. Plate III. Fig. 5.
If there be any number of magnitndes AB, CD, equimulti-
ples to a like number of magnitudes E, F, each to each ;
what ever multiple one magnitude AB is of one E, the fame
multiple is all the magnitudes ABCD to all the other mag-
nitudes E+F.
Let AG, GH, HB, the parts of the quantity AB, be
equal to E, and alſo let CI, IK, KD, the parts of the
quantity CD, be equal to F. The Number of theſe are
put equal to thoſe. Now whereas AG+CI (a) =E+
F
a 2. ex.
64
The fifth Book of
2 2. ax.
a 2. ax.
a byp.
b 2. 5.
c 2. 5.
F: (a) and GH+IK=E+F; (a) and HB+KD=E
+F, it is evident that AB+CD doth fo often contain
E+F as one AB contains E, Which was to be done.
PROP. II. Plate III. Fig. 6.
If the firft AB be the fame multiple of the fecond C, as the
third DE is of the fourth F, and if the fifth BG be the fame
multiple of the fecond C, as the fixth EH is of the fourth F;
then jall the first and fifth taken together (AG) be the lame
multiple of the fecond C, as the third and fixth taken together
(DH) is of the fourth F.
The number of parts in AB equal each to C is put
equal to the number of parts in DE, whereof each part
is equal to F. Likewife the number of parts BG is put
equal to the number of parts in EH. Therefore the
number of parts in AB+BG is equal to the number of
parts in DELEH. (a) That is, the whole line AG is
the fame multiple of C, as the whole line DH is of F.
Which was to be demonftrated.
PRO P. III. Fig. 7.
If the first A be the fame multiple of the fecond B, as the
third C is of the fourth D, and there be taken EI, FM equi-
multiples of the firft and third, then will each of the magni-
tudes taken be equimultiples, the one EI of the fecond B, the
the other FM of the fourth D.
Let EG, GH, HI, the parts of the multiple EI, be equal
to A; alfo let FK, KL, LM, the parts of the multiple
FM be equal to C; (a) the number of theſe is equal to
the number of thoſe. Moreover A (that is) EG or GH
or HI, is put the fame multiple of B, as C, or FK, &c.
is of D. (6) Therefore EG+GH is the fame multiple
of the ſecond B, as FK+KL is of the fourth D. (c) By
the fame way of arguing is EI (EH+HI) the fame mul-
tiple of B, as FM (FL+LM) is of D.´ Which was to
be demonftrated.
PRO P. IV. Fig. 8.
If the first A have the fame ratio to the ſecond B, as the
third C to the fourth D; then alfo E and F the equimultiples
of the first A and the third C, ſhall have the ſame ratio to
G and H, the equimultiples of the fecond B and the fourth
D, according to any multiplication, if ſo taken as they answer
each to other (EGF: H.)
Take
EUCLIDE'S Elements.
65
:
Take I and K equimultiples of E and F; and alfo L
and M equimultiples of G and H. (a) Then is I the fame
multiple of A, as K is of C; (a) and alfo L is the fame
multiple of B, as M of D. Therefore whereas it is
A: B(6) C: D; according to the fixth definition, if
I be L, then confequently after the fame
manner is K‚—‚—M, Therefore when I and K
are taken the fame multiples of E and F, as L and M of
G, and H, then will it be by the feventh definition E: G
F: H. Which was to be demonftrated.
C
Coroll:
From hence is generally demonftrated the proof of inverfe ratio.
For becauſe A:B::C: D, therefore if E,
G, then is (c) likewiſe F —,=, H; therefore
it is evident, that if G C,
E, then is HC;=,
F; (d) therefore B: A:: D: C. Which was to be
demonftrated.
PRO P. V. Plate III. Fig. 9.
If a magnitude AB be the fame multiple of a magnitude
CD, as a part taken from the one AE is of a part taken from
the other CF; the refidue of the one EB, ſhall be the ſame
multiple of the refidue of the other FD as the whole AB is
of the whole CD.
Take another GA, which ſhall be the fame multiple
of FD, the refidue, as AB is of the whole CD, or as
the part taken away AE, is of the part taken away
CF. (a) Therefore the whole GA AE is the fame
multiple of the whole CF+FD, as the one AE is of
the one CF, that is, as AB is of CD; therefore GE (b)=
AB; and (c) ſo AE, which is common, being taken
away, there remains GA-EB. Therefore, &c.
་་་་་་་་
PROP. VI. Fig. io.
If two magnitudes AB, CD, are equimultiples of two
magnitudes E, F; and fome magnitudes AG and CH equi-
multiples of the fame E, F, be taken away; then the refidues
GB, HD, are either equal to these magnitudes E, F, or else
tquimultiples of them.
For becauſe the number of parts in AB, whereof each
is equal to E, is put equal to the number of parts in
CD, whereof each is equal to F; and alfo the number of
E
parts
a 3. 5.
b hyp:
ċ 2. def. 5:
a 6: def: 5:
a 1.
5:
b 6. axe
C 3. at:
66
The fifth Book of
23. ax.
a 6. ax.
b6. def. 5.
e cor. 4. 5.
parts in AG equal to the number of parts in CH; If from
one you take AG, and from the other CH, (a) there re-
mains the number of parts in the remainder GB equal to
the number of parts in HD; therefore if GB be once E,
then is HB once C; if GB be many times E, then is HD
fo many times C. Which was to be demonftrated.
PROP.
VII. Plate III. Fig. 11.
Equal magnitudes A and B have the fame proportion or
ratio to the fame magnitude C. And one the jame magnitude
C hath the fame ratio to equal magnitudes A and B.
Take D and E equimultiples of the equal magnitudes
A and B, and F any multiple of C; then is D (a) — E.
Wherefore if D
Wherefore if DC,=,F, then alfo E will be
C',=, ~F; (b) therefore A: C : : B:C; and (c) by
invcrfion C: A :: (c) C: B. Which was to be demonſtrated.
Schol.
If instead of the multiple F, two equimultiples be
taken, it will be the fame way proved that equal mag-
nitudes have the fame ratio to any other magnitudes that
are equal between themſelves.
PRO P. VIII. Fig. 12.
Of unequal magnitudes AB, AC, the greater A B bath
a greater ratio to the fame third line D, than the leſſer AC;
and the fame third line D hath a greater ratio to the leſſer
AC, than to a greater AB.
Take EF, LG equimultiples of the faid AB, AC, fo
that EH, a multiple of D, may be greater than EG, but
leffer than EF, (which will eaſily happen, if both EG
and GF be taken greater than D.) It is manieft from 8
AB
def. 5. that
D
AC
D
Ꭰ
and
D
AB
AC
Which was to be demonftrated.
PROP. IX. Fig. 13.
Magnitudes, which to one and the fame magnitude have the
fame ratio, are equal the one to the other.
And if a magni-
tude have the fame ratio to other magnitudes, thofe magnitudes
☛re equal one to the other.
1. Hyp.
EUCLIDE'S Elements.
67
B
a 8. 5.
t. Hyp. If A: C:: B: C; I fay that AB. For
A
let A be greater or lefs than C; (a) then is-or
Which is contrary to the Hypothefis.
C
2. Hyp. If C: B::C: A. I fay that AB. For let
C
A be B, (b) then
pothefis.
B
C
A
C Which is against the Hy-
PROP. X. Plate III. Fig. 14.
Of magnitudes having ratio to the fame magnitude, that
which has the greater ratio, is the greater magnitude; and
that magnitude to which the fame bears a greater ratio, is
the lefer magnitude:
A
B
1. Hyp: If I fay that AB.
i.
--
C
C
For if it be faid that AB. (a)then A:C::B: C. Which
is contrary to the Hypothefis. If AB, (b) then is
A
10
C
B
Which is alſo against the Hypothefis:
b 8.5:
a 7. 5.
b 8. 5.
C
C C
2. Hyp. If
I fay that BA; for if
B A
C 7.5°
d 8.5°
you fay BA, it's against the Hypothefis, for it will
(c) follow that C: B::C: A. If you fay BA, (d) then
C
is
با
C
A B
Which is alfo aginft the Hypothefis.
PROP. XI. Fig. 15, 16, 17:
Proportions which are one and the fame to any third, are
alfo the fame one to another:
Let A:B::E: F, and CD:: E: F. I fay that A: B
::C: D. Take G, H, I, equimultiples of A, C, E;
and K, L, M, equimultiples of B, D, F. Now (a) be- a hyp.
caufe ABE: F; if Ĝ,=, K, (b) then after
G
the fame manner I,=, M. And likewife (a)
becauſe E: FC: D, if I,,M, (b) then is
H likewiſe,,L() wherefore A: BC: D.
Which was to be demonftrated.
E 2
Scholi
b 6. def. 5
c6.def
c 6. def 5
68
The fifth Book of
a 1. 5.
b byp.
Schol.
Proportions that are one and the fame to the fame
proportions, are the fame betwixt themſelves.
PROP. XII. Plate III. Fig. 15, 16, 17.
If any number of magnitudes A, B, C, D, E and F be
proportional; as one of the antecedents A, is to one of the con-
fequents B, fo are all the antecedents A, C, E, to all the
confequents, B, D, F.
Take G, H, I, equimultiples of the antecedents, and
K, L, M, of the confequents. Becauſe that one G is
the fame multiple of one A, (a) as all G, H, I, are of
all A, C, E; and becauſe one K is the fame multiple
of one B, as all K, L, M, are of all B, D, F: More-
over becauſe A : B (b) : : C: D (b) : : E : F, if G be,
=> or K, then will H likewiſe be,, ~ L,
and I,=, ~M; and fo if G —, —, — K, in
like manner will G+H+ I be ‚—‚—K+L+M;
c 6. def. 5. (c) wherefore A: BA+C+E: B+D+F. Which
was to be demonftrated.
26. def. 5.
Corol.
,
Hence, if like proportionals be added to like pro-
portionals, the wholes ſhall be proportional.
PRO P. XIII. Fig. 15, 16 17.
If the first A have the fame ratio to the fecond B, that
the third Chath to the fourth D, and if the third C have
a greater proportion to the fourth D, than the fifth E to
the fixth F; then alfo fhall the first A have a greater
proportion to the fecond B, than the fifth E to the fixth F.
Take G, H, I, equimultiples of A, C, E, and K, L,
M, equimultiples of B, D, F. Now becauſe that A:
BC:D, if HCL, (a) then is GK; but becauſe
C
b 8. def. 5.
D
c 8 def. 5.
E
. F
(6) it may be that HL, and yet I not —
M. (c) Therefore
A
E
Which was to be demon.
B
F
Schol,
EUCLIDE's Elements.
69
Schol.
C
E
A
E
But if
then alſo is
Alſo, if
>
D
F
B F
D
ALAUTA
C
E
A
E
A
then is
با
And if
D
F
B
F
B
E
A
E
then is
F
B
F
PROP. XIV. Plate III. Fig. 18.
If the frft A have the fame ratio to the fecond B, that
the third C hath to the fourth D; and if the first A be
greater than the third C; then fall the fecond В be greater
than the fourth D. But if the first A be equal to the third
C, then the fecond В fhall be equal to the fourth D; but if
A be lefs, then is В alſo lefs.
Let A C; (a) then
II
с с
(c) therefore
D
B
A
C
A
C
(b) but
;
B
B
BD
a 8. 5.
b hyp.
c 13. 5.
d 10. 5.
e 7.5.
; (c) therefore BD. By the
like way of argument, if A → C, (a) then is B¬D.
But if A be put equal to C, then C: B: (e) A: B (f)
:: C: D. (g) Therefore BD. Which was to be
demonftrated.
Schol.
A
By an argument à fortiori, if
B
C
~~, and A □ C,
AC,
D
then is BD. Likewife if A=B, then is CD, and
if A, or B, then alfo is C or
PROP. XV. Fig. 19.
D.
Parts C and F are in the fame ratio, with their like mul-
tiples AB and DE, if taken correfpondently. (AB: DE: :
C: F.)
Let AG, GB, parts of the multiple AB, be equal to
C; and let DH, HE, parts of the multiple DE, be equal
to F. (a) The number of theſe parts is equal to the num-
E 3
ber
f hyp.
gii. 5.&
9. 5.
a hyp.
1
70
The fifth Book of
b 7.5:
C 12. 5:
a 15. 5.
b hyp.
C
11.5. &
i 14.5.
d 6. def. 5.
a 1.5
b conftr.
CI. 5.
d 2. 5.
ber of thoſe.
Therefore whereas
AG:C::
Therefore whereas (b) AG: C:: DH: F,
and GB: CHE: F; therefore is (c) AG+GB (AB)
:DH+HE (DE): C: F. Which was to be demonftrated.
:
PRO P. XVI. Plate III. Fig. 20.
If four magnitudes A, B, C, D, are proportional, they
alſo ſhall be alternately proportional (A : С : : B : D)
Take E and F equimultiples of A and B; take alfo G
and H equimultiples of C and D. Therefore E: F (a)
: AB(6)::C: D (a) : : G: H. Wherefore if E
C‚—‚—(‹)G, then likewife is F,, H. (d)
Therefore A: C::B:D. Which was to be demonftrated.
•
Schol.
>
Alternate ratio has place only when the quantities are
of the fame kind. For heterogeneous quantities are not
compared together.
PRO P. XVII. Fig. 21.
If magnitudes compounded are proportional (AB: CB::
DF: FE.) they ſhall be proportional alſo when divided.
(AC: CB: DF : FE.)
Take GH, HL, IK, KM, in order equimultiples of
AC, CB, DF, FE; and alſo LN, MO, equimultiples
of CE, FE. The whole GL is (a) the fame multiple of
the whole AB, as one GH is of one AC; (b) that is, as
IK of DF, (c) or as the whole IM of the whole DE.
Alfo HN (HL+LN) is (d) the fame multiple of CB,
as KO, (KM+MO) is of FE. Therefore, whereas by
Hyp. AB BC: DE: EF, if GL be,,
:
HN,
then likewife (e) will IM —, —, — KO. Take from
e 6. def. 5. thefe HL, KM, that are equal; and if the remainder
GH be,, LN, (ƒ) then will IK, =, a
MO; (g) whence AC: CB :: DF, FE. Which was to
be demonftrated.
f
5. ax.
g 6. def. 5.
a 17.5.
b typ. &
11. 5.
C 14 5-
dg. ak.
PROP. XVIII. Fig. 22.
If magnitudes divided are proportional (AB: BC: : DE
: EF,) the fame alſo being compounded ſhall be proportional
(AC: CB :: DF : FE.)
For if it can be, let AC: CB:: DF FG FE. (a)
Then by diviñon will AB: BC:: DG:GF; (b) that is,
DG: GF :: DE. EF; and fince DGDE, (c) there-
fore is GF EF. Which is abfurd. The like abfurdity
will follow if it be faid AC: CB:: DF: GFFE (d).
PROP.
EUCLIDE's Elements,
71
1
PRO P. XIX. Plate III. Fig. 23.
If the whole AB be to the whole DE as the part taken away
AC is to the part taken away DF, then hall the refidue CB
be to the refidue FE, as the whole AB is to the whole DE.
Becauſe (a) At: DE :: AC : DF (b) therefore by
permutation AB AC: DE: DF, (c) and thence by
divifion AC: CB :: DF : FE; (b) wherefore again by
permutation AC: DF:: CB: FE (d) that is, AB : DE
:: CB: FE. Which was to be demonftrated.
Coroll.
Hence, If like proportionals be fubtracted from like
proportionals, the remainders fhall be proportional.
2. Hence is converſe ratio demonftrated.
Let AB CB:: DE: FE. I fay that AB: AC :
DE: DF. For by (a) permutation AB: DE :: CB : FE,
(b) therefore AB: DE: AC: DF, whence again by
permutation AB: AC:: DE: DF. Which was to be
demonftrated.
PRO P. XX. Fig. 24.
If there are three magnitudes A, B, C, and others D, E, F,
equal to thofe in number, which being taken two and two in
each order are in the fame ratio, (A:B:: D:E; and B: C
:: E: F,) and if ex æquo the firſt ▲ be greater than the
third C; then hall the fourth D be greater than the fixth
F. But if the firft A be equal to the third C, then the
fourth Dhall be equal to the fixth F; and if A be lefs
than C, then hall ▷ be leſs than F.
1. Hyp. If AC. Becaufe (a) E: F:: B: C, by (b) in-
verfion it ſhall be F:E::C: B. (c) But
F
A D
fore
E
C A
(d)there-
B
B
or -, (e) therefore DF. Which was
B E
to be demonftrated.
2. Hyp. By the fame way of arguing, if AC, it
will appear that D ~ F.
3. Hyp. If A=C. Becauſe F:EC: B: (f) A: B
:: D: E, (g) therefore is D=F.
Which was to be dem.
PRO P. XXI.
Fig. 24.
If there are three magnitudes A, B, C, and others alfo
D, E, F, equal to them in number, which taken two and two
E 4
are
a hyp.
b 16.5″
C 17. 5.
d hyp. &
11.5.
5.
a 16.
b 19. 5.
a hyp.
b cor. 4. 5.
c hyp, and
8. 5.
dſchol.13.5
e 10. 5.
£7.5.
g 11.5.&
9. 5.
72
The fifth Book of
+
a byp.
b 8.5.
are in the fame ratio; and their proportion pertubate (A : B
::E: F, and B: C: :D: E.) and if ex æquo the firft
A be greater than the third C, then is the fourth D greater
than the fixth F; but if the first be equal to the third, then is
the fourth equal to the fixth; if lefs, fo is the other likewife.
1. Hyp. If ACC; then becauſe (a) D: E:: B: C,
A
C
therefore inverſely E: D::C: B, but (6)
E
c
fch. 13. 5. (c) therefore
(c) therefore →→
d 10. 5.
D
DEF.
;
B
B
A
E
that is, than (d) therefore
B
F
e 7.5.
f hyp.
8 9: 5.
a hyp.
b
4. 5.
C 20. 5.
d 6. def. 5.
a 15. 5.
b byp.
€ 4.5.
2. Hyp. By the like argument, if AC, then iş
DF.
3. Hyp. If A=C; then becauſe E: D:: (e) C: B
: (e) A: B:: (ƒ) E : F, (g) therefore is DF,
Which was to be demonftrated.
PROP. XXII. Plate III. Fig. 25.
If there be any number of magnitudes A, B, C, and others
equal to them in number D, E, F, which taken two and two
are in the fame ratio (A: B::D: E and B: C
B:C::E:F)
they shall be in the fame ratio alfo by equality (A : C:: D:F.)
Take G, H, equimultiples of A, D; and I, K, of B, E;
and alfo L, M, of E, F.
H:
Becauſe (a) A: B :: D: E, (b) therefore G : I
K; and in like manner I: L::K: M. therefore if G
C,=, ~ 1, (c) then is H‚—‚— M ; (d) there-
L, HC,
fore A: CD: F. By the fame way of demonftration
if further C: N: : F: O, then by equality A : N
D:0. Which was to be demonftrated.
PROP. XXIII. Fig. 26.
If there are three magnitudes A, B, C, and others D, E,
F, equal to them in number, which taken two and two are
in the fame ratio, and their proportion perturbate (A : B :
E: F, and BC: D: E) they shall be in the fame ratiq
alfo by equality (A: C: D:F)
:
Take G, H, I, equimultiples of A, B, D; and alfo
K, L, M, equimultiples of C, E, F. Then G: H:: (a)
AB (6) E F (a): L: M. Moreover becauſe (b)
DE, thence is (c) H: K:: I: L; therefore
ØE
B: C:
:
G, H, K, and I, L, M, are as in 21. 5. Therefore if
G
EUCLID E's Elements.
73
G be,=,K, then is likewife I, -, ~ M,
and fo (d) confequently A: C:: D:F. Which was to
be demonftrated.
If there are more magnitudes than three, this way of de-
monſtration holds good in them also.
Coroll.
From hence it follows, that ratio's compounded of
the fame ratio's, are among themſelves the fame; as al-
fo that the fame parts of the fame ratio's, are among
themſelves the fame.
PRO P. XXIV. Plate III. Fig. 27.
If the first magnitude AB, has the fame ratio to the fecond
C, which the third DE, has to the fourth F; and if the
fifth BG has the fame ratio to the fecond C, which the fixth
EH has to the fourth F; then shall the first compounded with
the fifth (AG) have the fame ratio to the fecond C, which
the third compounded with the fixth (DH) has to the fourth F.
For becauſe (a) AB : C :: DE: F, and by the Hyp.
and inverſion C: BG :: F: EH; therefore by (b) equa-
lity AB BG: DE: EH, whence by compounding,
AG: BG:: DH: EH. Alfo (c) BG: C: : EH: F.
Therefore again by (b) equality AG C DH: F.
Which was to be demonftrated.
:
PROP, XXV. Fig. 28.
:
:
If four magnitudes are proportional (AB CD E:F)
the greatest AB and the leaft F ſhall be greater than the reſt
C, D, and E.
-
...
Make AGE, and CHF. Becauſe AB: CD:
(a) E: F: (b)AG: CH, (c) thence is AB: CD: GB
: HD; (d) but AB CD, (e) therefore GB HD.
But AG+FE+CH, therefore AG+F+GB ← E
+GBE
+CH+HD; that is, AB+FE+CD. Which was
to be demonftrated.
$
Thefe propofitions which follow are not Euclide's, but
taken out of other Authors, and are here ſubjoyned be-
caufe of their frequent ufe.
PROP. XXVI. Fig. 29.
If the first have a greater proportion to the fecond, than the
third to the fourth; then by converfion, the fecond shall have
a less proportion to the first, than the fourth to the third.
1
Let
d 6. def. 5.
• 228 23;
5. and 20.
def. 1.
a hyp.
b 22. 5:
c byp.
a hyp.
b 7. 5.
c 19. 5.
d hyp.
e fch.14.5.
74
The fifth Book of
A
C
Let
>
B
D
CE
I ſay that
A E
B
A
D
コ
​For conceive
C
; (a) therefore——- ; (b)whenceACE; (c)there-
BB
D B
a 13. 5.
B
B
D
fore
~~, (d) or -.
Which was so be demonftrated.
A
E
C
bio. 5.
c 8. 5.
d cor. 4. 5.
PROP. XXVII. Plate III. Fig. 29.
If the firft have a greater proportion to the fecond, than
the third to the fourth; then alternately the firſt ſhall have a
greater proportion to the third, than the fecond to the fourth.
A
C
A B
Let ご
​then I fay
For conceive
B
D
C D
E
C
A
E
(a) therefore AE, (6) and therefore-
a 10. 5.
B
D
C
C
b 1. 5.
B
(c) or
£ 16.5.
D
Which was to be demonftrated.
PROP. XXVIII. Fig. 30.
If the first bave a greater proportion to the ſecond than
the third to the fourth, then the first compounded with the
fecond ball have a greater proportion to the fecond, than the
third compounded with the fourth, to the fourth.
AB
DE
AC
DF
Let
I ſay that
For
BC
EF
BC
EF
GB
DE
conceive
a 10. 5.
BC
EF
b
4. ax.
, (a) therefore is, ABGB, add
BC to each part, then (6) will ACGC; (c) therefore
€ 8. 5.
AC
GC
DF
d 18. 5.
>
(d) that is,
Which was to be dem.
BC
BC
EF
PROP. XXIX. Fig. 30.
If the first compounded with the fecond has a greater
proportion to the ſecond, than the third compounded with
the
EUCLIDE's Elements,
75
the fourth hath to the fourth; then by divifion the first
fhall have a greater proportion to the fecond, than the third to
the fourth.
AC
DF
AB
ᎠᏴ
Let
then I fay
C
For
BC
EF
BC
EF
GC
DF
conceive
(a) therefore ACGC. Take
a 10, 5.
BC EF
away BC, which is common, there (b) remains ABC
AB
GB
DE
GB; (c) therefore
(d) or
Which
BC
BC
EF
5.
ъ ax.
c 8. 4.
d 17: 5:
avas to be demonftrated.
PROP. XXX, Plate III. Fig. 31.
If the first compounded with the fecond, has a greater
proportion to the fecond, than the third compounded with
the fourth, bath to the fourth; then by converfe ratio ſhall
the first compounded with the fecond have a leffer ratio to
the first, than the third compounded with the fourth shall
have to the third.
AC
DF
AC DF
Let
Then I fay that
BC
EF
AB
DE
AC
DF
For becauſe that-
-(a)
(b)therefore, by divifion,
BC
EF
a byp.
b 29. 5.
AB
DE
by converſion (c) therefore
BC
EF
BC EF
・コ
​AB DE
c 26.
5.
AC
DF
and (d) therefore by compoſition
C
Which d 28. 5:
AB
DE
was to be demonftrated.
PROP. XXXI. Fig. 32,
If there are three magnitudes A, B, C, and others also
D, E, F, equal to them in number; and if there be a
greater proportion of the firft of the former to the fi
cond, than there is of the first of the last to the fecond
(一​)
) and there be alſo a greater proportion of
the fecond of the firft magnitudes to their third, than there
is
7.6
The fifth Book of
is of the fecond of the last magnitudes to their third
B E
(C-F.
)
Then by equality alfo fhall the ratio of the
firft of the former magnitudes to the third, be greater than
the ratio of the first of the latter magnitudes, to the third
A
D
F.
GE
Conceive
, (a) then is BG, and (b) there-
a 10. 5.
b 8. 5
in in
C
F
A A
H D
fore
€ 13. 5.
Again conceive
(c) then
G
B,
GE
H
A
H
A
d 10. 5.
, therefore much more
- —, (d) wherefore
G B
G G
A
H
D
e 8. 2.
AH, (e) and confequently-
-(f) or
W.W.D.
£ 22. 5.
C
F
a 10. 5.
PROP. XXXII. Plate III. Fig. 32.
If there be three magnitudes A, B, C, and others
D, E, F, equal to them in number; and there be a
greater proportion of the first of the former magnitudes to
the fecond, than there is of the jecond of the latter to
the third
(
A
B
E
F
-)
and alfo the ratio of the fe-
cond of the former to the third be greater than the ratio
of the first of the
latter to the ſecond (
B
D
C
E.
}
then by equality alfo fhall the proportion of the first of
the former to the third, be greater than that of the first
A
of the latter to the third ( (~-
Suppofe
G D
A A
10 n
b 8. 5.
c ſchol.
13. 5.
(b) —
G
CHIO
CE
C
D
:)
.)
F.
then is (a) B☛ G, and therefore
-. Again, Suppoſe
B
A
G
H
E
; therefore is (c)
G
F
II
, and confequently (a) A H, and thence (6)
A
EUCLIDE's Elements.
77
410
A H
D
(d) or
Which was to be demonftrated.
d 13. 5.
C
C
F
PROP. XXXIII, Plate III. Fig. 33.
If the proportion of the whole AB to the whole CD be
greater than the proportion of the part taken away AE to the
part taken away CF; then shall also the ratio of the re-
mainder EB to the remainder FD be greater than that of
the whole AB to the whole CD.
AB
AE
Becauſe that
(a) =
(b) therefore by permu-
a hyp.
CD
CF,
b 27. 13
AB
CD
AB
tation
>
(c) therefore by converfe ratio
c 30. 5.
AE
CF
EB
CD
AB
EB
, and by permutation again
n
FD
CD
FD
Which was to be demonftrated.
PRO P. XXXIV.
If there be any number of magnitudes, and others alfo
equal to them in number; and the proportion of the first of
the former to the first of the latter be greater than that of
the fecond to the fecond, and that greater than the proportion
of the third to the third, and fo forward: all the former
magnitudes together shall have a greater ratio to all the
latter together, than all the former, leaving out the first,
fball have to the latter, leaving out the firft; but less than
that of the first of the former to the first of the latter; and
laftly, greater than that of the last of the former to the last
of the latter.
You may pleaſe to confult Interpreters for the demon-
ftration hereof, we having for brevity fake omitted it,
and becauſe 'tis of no uſe in thefe Elements.
The End of the fifth Book.
The
[ 78 ]
The SIXTH BOOK
O F
1
E U
UCLIDE's
Plate III. I.
Fig. 33.
Fig. 34.
Fig. 23.
Fig. 33.
ELEMENTS.
L
Definitions.
Ike right-lined figures (ABC, DCE) are
fuch whoſe ſeveral angles are equal one to
the other, and alfo their fides about the
equal angles, proportional.
The Angle BDCE, and AB: BC :: DC CE.
Alfo the angle AD, and BA : AC:: CD: DE. Laftly
the angle ACBE, and BC: CA :: CE: ED.
II. Reciprocal figures are (BD, BF) when in each of
the figures there are terms both antecedent and confe-
quent (that is, AB: BG:: EB : BC.
III. A right-line AB is faid to be cut according to
extreme and mean proportion, when as the whole AB
is to the greater fegment AC, fo is the greater fegment
AC to the lefs CB (AB: AC :: AC: CB.)
IV. The altitude of any figure ABC, is a perpendicu
lar line AG, drawn from the top A, to the baſe BC.
V. A ratio is faid to be compounded of ratio's, when'
the quantities of the ratio's, being multiplied into one
another, produce a ratio, As the ratio of A to C is
A B
compounded of the ratio's of A to B and B to C. For - X-
B C
AB
A
a 20. def. 5.
(a) ==
(6)=
b 15. 5.
BC
C.
PROP. I. Plate III. Fig. 35.
Triangles ABC, ACD, and parallelograms BCAE,
CDFA, which have the fame height, are to each other,
as their bafes, BC, CD.
(a) Take
EUCLIDE's Elements.
79
(a) Take as many as you pleaſe, BG, GH, equal to
BC, and alfo DI-CD, and join AG, AH, AI.
(b) The triangles ACB, ABG, AGH, are equal, and
(b) alſo the triangle ACD ADI. Therefore the triangle
ACH is the fame multiple of the triangle ACB, as the
bafe HC is of the bafe BC; and the triangle ACI the
fame multiple of the triangle ACD, as the bafe CI is of
CD. But if HC
But if HC,, CI, (c) then is likewife the
triangle AHC, E, ACI; and (d) therefore BC :
CD the triangle ABC: ACD:: (e) Pgr. CE: CF.
Which was to be demonftrated.
Schol. Plate III, Fig. 37.
Hence triangles, ABC, DEF, and Pgrs. AGBC, DEFH,
whofe bafes BC, EF are equal, are to each other as their
altitudes, AI, DK.
a 3. I.
b 38. I.
cfch, 38.1.
d6. def. 5.
e 41, t.&
15. 5.
a 3. I.
b 7.5.
C I. 6.
(a) Take IL CB, and KM EF; and join LA,
LG, MD, MH, then it is evident, that the triangle
ABC: DEF:: (6) ALI: DKM:: (c) AI: DK::(d)Pgr. d41. 1.&
AGBC: DEFH. Which was to be demonftrated.
PROP. II. Fig. 36.
If to one fide BC of a triangle ABC, be drawn a parallel
right-line DE, the fame shall cut the fides of the triangle pro-
portionally (AB: BD :: AE: EC.) And if the fides of
the triangle are cut proportionally (AD: BD :: AE: EC)
then a right-line DE, joining the points of a fection DE,
hall be parallel to BC, the other fide of the triangle.
Draw CD and BE.
1. Hyp. Becauſe the triangle DEB (a) =DEC, (b) there-
fore ſhall be the triangle ADE: DBE :: ADE: ECD.
But the triangle_AED: DBE :: (c) AD: DB, and the
triangle ADE: DEC :: AE: EC; (d) therefore AD :
DB:: AE: EC.
2. Hyp. Becauſe AD: DB:: AE: EC, (e) that is as
the triangle ADE : DBE :: ADE : ECD; (ƒ) therefore
is the triangle DBE ECD; and (g) therefore DE, BC
are parallels. Which was to be demonftrated.
Schol.
If there are drawn feveral lines DE, FG parallel to
one fide BC of a triangle, all the fegments of the
Aides fhall be proportional.
15. 5.
a 37. I.
b 7. 5.
C I. 6.
d11.5.
e 1. 6.
f 9. 5-
8 39. I.
For
1
80
The fixth Book of
ǎ 2.6.
a 5. 1.
b 32. 1.
c hyp.
d 27. 1.
€ 2.6.
f 2. 6.
g 29. 1.
h
5.1
k 1. ax.
For DF: FA (a): EG: GA; and compounding
and inverting, FA: DA :: GA : EA; (a) and DA : DỄ
:: EA: EC; therefore by equality DF:DB:: EG: EC.
Which was to be demonftrated:
Coroll.
If DF: DB:: EG: EC; (a) then BC, DE, FG,
fhall be parallels:
PROP. III. Plate III. Fig. 38.
If an angle BAC of a triangle BCA be bifected, and
the right-line AD, that bifects the angle, cut the base alſo;
then all the fegments of the base have the fame ratio that
the other fides of the triangle have, (BD: DC: : AB :
AC.) And if the fegments of the bafe have the fame ratio,
that the other fides of the triangle have (BD : DC :: AB :
AC) then a right-line AD drawn from the top A to the
fection D, fhall bifect that angle BAC of the triangle.
Produce BA, and make AE-AC, and join CE.
1. Hyp. Becauſe AE AC, therefore is the angle
ACE (a) = E (b) ≈ half BAC (c) = DAC ; (d) there-
fore DA, CE are parallels. (e) Wherefore BA : AE (AC)
:: BD: DC,
*
2. Hyp. Becauſe BA: AC (AE) :: BD: DC,(f) there-
fore are DA, CE parallels; and (g) therefore is the angle
BAD=E; and the angle DAC (g)=ACE (b) =E, (k)
therefore the angle BAD DAC. Wherefore the angle
BAC is bifected. Which was to be demonftrated.
PROP. IV. Fig. 39.
Of equiangular triangles ABC, DCE, the fides are pro-
portional which are about the equal angles, B, DCE, (AB
: BC: DC: CE, &c.) And the fides AB, DC, &c.
which are fubtended under the equal angles ACB, E, &c.
are homologous, or of like ratio.
Set the fide BC in a direct line to the fide CE, and
a 32. 1. & produce BA and ED till they (a) meet in F.
13. ax.
b hyp.
c 28, 1.
d
34.
6.2. 6.
I.
Becauſe the angle B (b) ECD, (c) therefore BF, CD
are parallel: Alio becaufe the angle BCA (b) = CED,
(e) therefore are CA, EF parallel. Therefore the figure
CAFD is a Pgr. (d) therefore AF=CD, and AC- (d)
FD. Whence it is evident, that AB AF (CD): : (e)
;
BC
EUCLIDE's Elements.
8 I
BC: CE. (f) By permutation therefore AB: BC: : CD:
CE, alfo BC: CE : : FD (AC) : DE. (ƒ) And thence by
permutation BC: AC::CE: DE. (g) Wherefore alfo
by equality AB: AC:: CD: DE. Therefore, &c.
Coroll
Hence AB: DC :: BC: CE :: AC: DE.
Schol.
Hence, if in a triangle FBE there be drawn AC, a
parallel to one fide FE, the triangle ABC fhall be
fimilar or like to the whole FBE.
PROP, V. Plate III. Fig. 40, 41.
If two triangles ABC, DEF, have their fides propor=
tional (AB: BC:: DE: EF, and AC: BC; : DF: EE,
and alfo AB AC :: DE: DF.) thoſe triangles are equian-
gular, and thofe angles equal, under which are fubtended the
homologous fides.
At the fide EF (a) make the angle FEG=B, and the
angle EFG C; (b) whence the angle GA. There-
fore GE EF (c) :: AB: BC :: (4) DE: EF. (e) And
BC::
therefore GE=DE. Likewife GF: FE (c) :: AC: C3
:: (d) DF: FE; (e) therefore GFDF. Therefore the
triangles DEF, GEF, are mutually equilateral._ (f)
Therefore the angle D=G=A, and the angle FED (ƒ)
=FEGB, and (g) confequently the angle DFE C.
Therefore, &c.
`
PROP. VI. Fig. 40, 41.
If two triangles ABC, DEF have one angle B equal to
one angle DEF, and the fides about the equal angles B,
DEF proportional (AB : BC :: DE: EF) then those trian-
gles ABC, DEF, are equiangular, and have thoſe angles equal,
under which are fubtended the homologous fides.
At the fide EF make the angle FEG = B, and the
angle EFG=C; (a) then will the angle G=A. There-
fore GE EF: (b) AB : BC: : (c) DE : EF, (d) and
therefore DE=GE. But the angle DEF (e) =В (ƒ)
=GEF; therefore the angle D (g) —G—A, (b) and
confequently the angle E FDC. Which was to be
demonftrated.
F
f 16.5.
g 22. 5.
a 23. 1.
b 32. I.
c 4.6.
d hyp.
e 11.5.&
9.5.
f 8. I.
g 32. I.
a 32. 1.
b
4. 6.
c hyp.
d
9. 5.
e hyp.
£ conftr.
g 4. I.
PROP.
h 32. 1.
82
The fixth Book of
a hyp.
b 32. 1.
C 4. 6.
d hyp.
PROP. VII.
Plate III. Fig. 43, 44.
If two triangles ABC, DEF have one angle A equal to
one angle D, and the fides about the other angles ABC, E,
proportional (AB: BC: DE: EF) and if they have the
remaining angles C, F, either both less or both greater than
a right-angle; then ſhall the triangles ABC, DEF, be equi-
angular, and have thoſe angles equal about which the pro-
portional fides are.
•
=
For, if it can be, let the angle ABC — E, and
make the angle ABGE. Therefore, whereas the
angle A (a)D, (b) thence is the angle AGB F.
Therefore AB BG (c) :: DE: EF: (d) AB: BC, (e)
:
therefore BG BC; (f) therefore the angle BGC=
BCG. (g) Therefore B G C, or C, is leſs than a right-
angle, and (b) confequently AGB or F is greater than a
right: Therefore the angles C and F are not of the fame
g cor. 17.1. fpecies or kind, which is againſt the Hypothefis.
e 9.5.
5. 1.
f
cor. 13. 1.
a hyp.
b 12. ax.
© 32. 1. &
4. 6.
PRO P. VIII. Fig. 42.
If a line AD be drawn from the right-angle A, of a
right-angled triangle ABC, perpendicular to the bafe BC;
then the triangles A DB, ADC, on each fide the perpendi-
cular, are fimilar both to the whole ABC, and to one
another.
For becauſe BAC, ADB are (a) right-angles, (b) and
fo equal, and B common; the triangles BAC, ADB, (c)
are like. By the fame way of arguing BAC, ADC,
are like; (d) whence alfo ADB, ADC will be like.
dvid. 21. 6. Which was to be demonſtrated.
e 1. def. 6.
23. 1.
b 31. 1.
b31.
Coroll.
Hence, 1. BD: DA (e) :: DA : DC.
2. BC: AC :: AC: DC, and CB: BA:: BA: BD.
PROP. IX. Fig. 45.
From a right-line given AB to cut off any part required,
as one third (AG.)
From the point A draw an infinite line AC any wiſe,
in which (a) take any three equal parts AD, DE, EF;
join FB, to which, from D, (b) draw the parallel DG,
and the thing is done.
For
EUCLIDE's Elements.
83
For GB: AG :: (c) FD: AD; whenće, by (d) com-
pofition, AB AG :: AF: AD; therefore fince AD=
one third of AF, AG fhall be one third of AB.
Which was to be done.
PROP. X. Plate III. Fig. 46.
To divide a given undivided right-line, AB (in F, Ĝ)
as another given right-line is divided (in D, E.)
Let a right-line BC join the extremities of the line
divided, and of the line not divided; and parallel to this,
from the points E, D, (a) draw EG, DF, meeting with
the right-line which is to be cut in G and F; then the
thing is done.
For let DH be (a) drawn parallel to AB. Then AD:
Then AD:
DE: (6) AF: FG,and DE: EC (6) :: DI; IH: : (©)
FG: GB. Which was to be done.
Schol. Fig. 47.
Hence we learn to cut a right-line given AB, into as many
equal parts as we please (Suppofe 5 ;) which will be more
caſily performed thus.
Draw an infinite line AD, and another BH parallel to
it, and infinite alfo. Of theſe take equal parts, AR, RS,
SV, VN; and BZ, ZX, XT, TL; in each line leſs
parts by one, then are required in A B; then let the
right-lines LR, TS, XV, ZÑ, be drawn; theſe lines fo
drawn fhall cut the right-line given AB into five equal
parts.
For RL, ST, VX, NZ, are (a) parallels; therefore
whereas AR, RS, SV, VN are (b) equal; (c) thence AM,
MO, OP, PQ, are equal alfo. Likewife, becauſe that
BZ = ZX, therefore is BQ=PQ, and therefore A B
is cut into five equal parts. Which was to be done.
PROP. XI. Fig. 48.
Two right-lines being given AB, AD, to find out a
third in proportion to them (DE.)
Join BD, and from AB, being produced, take BC÷
AĎ. Through C draw CE parallel to BD; with which
let AD produced meet in E, then is DE the proportional
required.
For AB: BC (AD) (@) :: AD: DE. Which was to
be done.
FA
c 2.6.
a 18. 5
Of
2 31. 1.
b 2. 6.
C 34. 1.&
7. 5.
a 33. to
b conftr
c 2. 6.
2. 6.
84
The fixth Book of
Or thus make the angle BDA (Fig. 42.) right, and
bi.cor.8.6. alfo the angle BAC right, then (6) BD: DA:: DA:
a 2.6.
a 31.3.
b cor. 8. 6.
a fch. 15. 1.
b 1. 6.
€ 7.5.
d 1. 6.
e II. 5.
DC.
PROP. XII. Plate III. Fig. 49.
Three right-lines being given, DE, EF, DG, to find out
a fourth proportional, GH.
1
Join E G, and through F draw FH parallel to E G ;
with which let DG, produced to H, meet. Then it is
evident that DE: EF (a) :: DG: GH. Which was
to be done.
PRO P. XIII. Fig. 50.
Two right-lines being given AE, EB, to find a mean
proportional, EF.
Upon the whole line A B, as a diameter, defcribe a
femicircle AF B, and from E erect a perpendicular FE
meeting with the periphery in F, then AE: EF:: EF:
EB. For let AF and FB be drawn; (a) then from the
right-angle of the right-angled triangle AFB is drawn
a right-line F E, perpendicular to the bafe. (6) There
fore AE FE: : FE EB. Which was to be done.
Coroll.
Hence, a right-line drawn in a circle from any point.
of a diameter, perpendicular to that diameter, and pro-
duced to the circumference, is a mean proportionalˇbe-
twixt the two fegments of that diameter.
PRO P. XIV. Fig. 34.
Equal Parallelograms BD, BF, having one angle ABC,
equal to one EBG, have the fides which are about the equal
angles reciprocal (AB: BG:: EB: BC;) and thoſe paral-
lelograms BD, BF, which have one angle ABC equal to one
EBG, and the fides which are about the equal angles reci-
procal, are equal.
For let the fides AB, BG, about the equal angles make
one right-line; (a) wherefore EB, BC, ſhall do the fame.
Let FG, DC, be produced till they meet.
1. Hyp. AB : BG (6) : : BD : BH: : (c) BF : BH :
(d) BE: BC, (8) therefore, &c.
2. Hyp•
EUCLIDE's Elements.
85
:
(g) BE BC
f1. 6.
2. Hyp. BD BH:: (f) AB: BG:: (g) BE : BC
(b) BF: BH. (k) Therefore the Pgr. BDBF.
Which was to be demonftrated.
PRO P. XV. Plate III. Fig. 51.
Equal triangles having one angle ABC, equal to one DBE,
their fides which are about the equal angles are reciprocal
(AB: BE::DB: BC.) And thofe triangles that have
one angle ABC equal to one DBE, and have also the fides
that are about the equal angles reciprocal (AB : BE: : DB :
BC) are equal.
Let the fides CB, BD, which are about the equal an-
gles be fet in a ſtrait-line; (a) therefore ABE is a right-
line. Let CE be drawn.
:
1. Hyp. AB: BE (b) the triangle ABC: CBE (c)
:: triangle DBE: CBE: () DB BC; (e) therefore, &c.
2. Hyp. The triangle ABC CBE :: (ƒ) AB : BE::
(3) DB: BC (b): the triangle DBE: CBE. (k) There-
fore the triangle ABC-DBE. Which was to be demon-
ftrated.
PROP. XVI. Fig. 52.
If four right-lines are proportional (AB: FG:: EF: CB)
the rectangle AC, comprehended under the extremes AB, CB,
is equal to the rectangle EG, comprehended under the means
FG, EF. And if the rectangle AC, comprehended under the
extremes AB, CB, be equal to the rectangle EG, compre-
hended under the means FG, EF, then are the four right-
lines proportional (AB: FG EF. CB.)
:
1. Hyp. The angles B and F are right, and (a) conſe-
quently equal, and by hypothefis AB: FG: EF: CB,
(b) therefore the rectangle AC-EG.
2. Hyp. The rectangle AC (c) EG, and the angle
B=F; (d) therefore AB: FG:: EF: CB. Which was
g hyp.
h 1. 6.
k 11. and
9. 5.
a ſch.15.1.
b 1.
6.
C 7.5.
di.
d 1. 6.
e 11. 5.
f 1.6.
g hyp.
h 1.6.
k 11. and
9. 5.
a 12.ax.
b 14.6.
c byp.
d 14. 6.
to be demonftrated.
Coroll.
Hence it is eaſy to apply a rectangle given EG to a
right-line given AB; (viz.) (e) by making AB EF:: Ee 12. 6]
FG: BC.
F 3
PRO P.
86
The fixth Book of
a hyp:
b 16. 6.
c 29. def. 1.
₫ hyp.
c 16.6,
@ 23: 19
b conftr.
€ 32, 1.
dz. ax.
e 4.6.
f 22. 5:
g1. def. 6.
PROP. XVII. Plate III. Fig. 52.
If three right-lines are proportional (AB: EF:: EF: CB)
the rectangle AC, made under the extremes AB, CR, is equal
to the fquare EG, made of the middle EF. And if the rect-
angle A C, comprehended under the extremes AR, CB, be
equal to the Square EG, made of the middle EF, then the
three lines are proportional, (AB: EF:: EF: CB.)
Take FG EF.
1. Hyp, AB: EF:: (a) EF (FG): CB, therefore the
rectangle AC (b) = EG (c) = EFq.
2. Hyp. The rectangle AC (d) to the fquare EG=
EFq; (e) therefore AB: EF:: FG (EF): BC. Which
was to be demonftrated.
Coroll.
Let AxB=Cq, then A: C:: C : B.
PRO P. XVIII. Fig. 53.
Upon a right-line given AB, to defcribe a right-lined figure
AGHB, fimilar and alike fituate to a right-lined figure
given CEFD,
Refolve the right-lined figure given into triangles; (a)
Make the angle ABHD, [a) and the angle BAH=
DCF, (a) and the angle AHG-CFE, (a) and the angle
HAGFCE, then AG HB, fhall be the right-lined
figure fought,
For the angle B (b) =D, and the angle BAH (6)=
DCF, (c) wherefore the angle AHB= CFD; (b) alfo
the angle HAG=FCE, and the angle AHG (b) =
CFE; (c) wherefore the angle G-E, and the whole
angle GAB (d) ECD, and the whole angle GAB (d)
:
:
EFD. The Polygons, therefore are mutually equian-
gular. Moreover becauſe the triangles are equiangular,
therefore AB BH (e) : : CD : DĚ; and AG: GH (e)
::CE: EF, Likewife AG AH :: (e) CE: CF, and
AH: AB:: CF: CD. (ƒ) From whence by equality
AG: AB::CE: CD, After the fame manner GH':
HB:: EF: FD. (g) Therefore the Polygons ABHG,
CDFE are fimilar and alike fituate. Which was to be done..
PROP. XIX. Fig. 54-
Like triangles ABC, DEF, are in duplicate ratio of their
bomologous fides, BC, EF.
(a) Let
EUCLIDE'S Elements.
87
(a) Let there be made BC: EF:: EF: BG, and
let A G be drawn. Becauſe that AB DE (b): :
BC: EF (c): EF: BG, and the angle B
:
DEF.
E, (d) therefore is the triangle ABG
But the triangle ABC: A BG: (e) B C BG, and (f)
BC BC
BGEF
twice. Which was to be demonftrated.
twice; therefore
ABC
ABG
ABC
that is,
DEF
=(8)
BC
EF
a 11. 6.
b cor. 4. 6.
c confir.
d 15. 6.
e 1.6.
f10.
f 10. def. 5.
I
g 11. 5.
Coroll.
Hence, If three right-lines (BC, EF, BG) are propor-
tional, then as the first is to the third, fo is a triangle
made upon the firſt BC, to a triangle fimilar and alike
deſcribed upon the fecond EF; or fo is a triangle de-
fcribed upon the fecond EF, to a triangle fimilar and
alike deſcribed upon the third.
PROP. XX. Plate III. Fig. 56, 57.
Like Polygons ABCDE, FGHIK,' are divided into equal
triangles ABC, FGH, and ACD, FHI, and ADE, FIK;
both equal in number and homologous to the wholes (ABC:
FGH :: ABCDEFGHIK: ACD: FHI: : ADE
FIK.) And the Polygons ABCDE, FGHIK, have a du-
plicate ratio one to the other of what one homologous fide BC
bath to the other homologous fide GH.
1 For the angle B (a) G, and AB: BC (a): :FG
: GH. (b) Therefore the triangles ABC, FGH, are equi-
angular. After the fame manner are the triangles AED,
FKI fimilar. Since therefore the angle BCA (6)=GHF,
and the angle ADE (6) FIK, and the whole angles
BCD, GHĨ, and the whole angles CDE, HIK are (c)
equal, there remains the angle ACD (a) = FHI, and the
angle ADC FIH; (e) from whence alfo the angle
CAD=HFI, therefore the triangles ACD, FHI are
fimilar. Therefore, &c.
a bip.
b 6. 6.
c hyp.
d
3. ax.
e 32. I.
2. Becauſe the triangles BC A, GHF are like, (f) f 19.
BCA BC
twice. For the fame reaſon is
therefore, BCA
CAD
6.
GHF GH
HFI
CD
DEA DE
twice; laftly
HI
-twice.
IKF IK
Now where-
:
:
as that B C G H (g): CD: HI (g):
therefore is the triangle BCA: GHF: CAD: HFI: :
DEA: IKF (4): the polygon ABCDEFGHIK:: BC h cor. 23.5.
Coroll k 12. 5.
: GH twice.
F4
DE: IK; (b)
8 byp. &
16.5.
88
The fixth Book of
* 18, 6,
a 1. def. 5.
a 19. 6.
b byp.
C 20. 6.
₫ cor. 23.5
Coroll.
1. Hence if there are three right-lines proportional,
then as the first is to the third, ſo is a polygon made
upon the first to a polygon made on the fecond fimilar
and alike deſcribed; or fo is a polygon made upon the
fecond, to a polygon made on the third fimilar and alike
defcribed.
Hence we have a method of inlarging or diminiſhing
any right-lined figure in a ratio given; For if you would
make a pentagon quintuple of that pentagon whereof
CD is the fide, then betwixt AB and 5 AB find out a
mean proportional, * upon this raiſe a pentagon like to
that given, and it fhall be quintuple of the pentagon
given.
2. Hence alſo, If the homologous fides of like figures
be known, then will the proportion of the figures be
evident, viz. by finding out a third porportional.
PRO P. XXI. Plate III. Fig. 58.
Right-lined figures ABC, DIE, which are fimilar to the
fame right-lined figure HFG, are alfo fimilar one to the
other.
=
— —
For the angle A (a)—H (a)=D; and the angle C (a)
G (a) E; and the angle B (a) = F (a) I. Alfo
(a) AB: AC::HF: HG:: DI: DE; and (a) AC:
CB: HG GF: DE: EI. And AB: BC:: HF
: FG:: DI; IE. Therefore (a) ABC, DIE, are fimi-
lar. Which was to be demonftrated.
PROP. XXII. Fig. 59.
If four right-lines are proportional (AB: CD:: EF:GH)
the right-lined figures alfo defcribed upon them being fimilar
and alike fituate, fhall be proportional (ABI: ČDK: :
EM: GO.) And if the right-lined figures defcribed upon
the lines, fimilar and alike fituate, be proportional (ABI:
CDK: EM: GO) then the right-lines alfo fhall be propor-
tional (AB : CD :: EF: GH.)
ABI
A B
=
EF
EM
1. Hyp. CDK (a) CDtwice(a)GHtwice (a)=GO
(b) therefore ABI: CDK:: EM : GO .
AB
ABI
2. Hyp. CD twice(a)=CDK
twice.
EM
(b) =GO
E F
= (c)GH.
Therefore (d) AB: CD:: EF: GH, Which
was to be demonftrated
Schol.
EUCLIDE's Elements,
89
Schol.
/
Hence is deduced the manner and reafon of multiplying
Jurd quantities, ex. g. Let 5 be to be multiplied into
3. I fay that the product will be 15. For by the
definition of multiplication it ought to be, as 1: √ 3
✔ 5, to the product. Therefore by this q. 1: q.
3:q1 5:q. of the product. That is, 1 : 3 :: 5, to the
fquare of the product, therefore the fquare of the product
is 15. Wherefore 15 is the product of √3 into
√ 5. Which was to be demonftrated.
THE ORE M. Plate III. Fig. 50.
If a right-line AB be cut any-wife in E, the rectangle
comprehended under the parts AE, EB, is a mean propor-
tional betwixt their fquares. Like wife the rectangle compre-
hended under the whole AB, and one part AE, or EB is a
mean proportional betwixt the fquare of the whole AB and
the Square of the faid part, AE, or EB,
Upon the diameter A B defcribe a femicircle; from
E erect a perpenpicular EF, meeting with the periphe-
ry in F, join AF, BF.
It's evident that AE: EF (a): : EF: EB, (b) therefore
AEq: EFq.:: EFq: EBq. (c) that is, AEq: AEB::
AEB: EBq. Which was to be demonftrated.
Moreover BA : AF :: (d) AF: AE, (e) therefore BAq.
AFq: AFq.: AEq. (f) that is, BAq: BAE:: BAE:
AEq. After the fame manner ABq.: ABE:: ABE:
BEq. Which was to be demonftrated.
Or thus: fuppofe ZA+F. It is manifeft that Aq.
:AF: (a) A: F: (a) AF: Fq. alfo Zq.: ZA:: (a)
A::ZA: Aq, and Zq: ZF :: (a) Z: F:: ZF: Fq.
Zi
•
PROP. XXIII.
Fig. 34.
Equiangular parallelograms AC, BF, have the ratio one to the
AC AB, BC
other, which is compounded of their fides. (AG=AG+
BF BG BE
Let the fides about the equal angles, B be (a) ſet in a
direct line, and let the Pgr. BH be compleated. Then is
AC AC BH
AB BC
Pet. Herig.
a cor. 8.6.
b22.6.
d cor. 8. 6.
c 17.6.
e 22. 6.
f 17. 6.
2 1.6.
a ſch. 19.
the ratio of BFBH+B=BG +BE Which b20 def.5.
was to be demonftrated.
C I. 6.
Coroll.
90
The fixth Book of
Andr.
Tacq. 15.5
*
*
35. 1.
14.6.
and 1.6.
29. 1.
b 4. 6.
C 22.
Coroll. Plate III. Fig. 61.
Hence, and from 34. 1. it appears, 1. That triangles
which have one angle equal (as at C) have a ratio compound-
ed of the ratio's cf the right-lines, (A to B, and C to F,)
containing the equal angle.
C
*
2. That all rectangles, and confequently all paralle-
iograms, have their ratio one to the other compounded of
the ratio's of bafe to baſe, and altitude to altitude. After
the like manner you may argue in triangles.
3. From hence is apparent how to give the proportion
of triangles and parallelograms.
Let there be two Pgrs. X and Z, whoſe baſes are AC,
CB, and altitudes CL, CF. Make CL: CF::CB: 0,
* then will it be X: Z:: AC:O.
PROP. XXIV. Fig. 55.
In every parallelogram ABCD, the parallelograms EG,
HF, which are about the diameter AC, are like to the
whole, and alfo one to the other.
For the Pgrs. EG, HF, have each of them one an-
gle common with the whole; (a) therefore they are equi-
angular to the whole, and alfo one to the other.
Alfo the triangles ABC, AEI, IHC (a) and the trian-
gles ADC, AĞI, IFC are equiangular mutually; (b)
therefore AE: EI:: AB :BC, and (b) AE : AI:: AB:
5. AC, and (b) AI: AG:: AC :: AD, (c) Therefore by e-
quality, AE: AG: AB: AD. (d) Therefore the Pgrs.
ÉG, BD are fimilar, After the fame manner are HF,
BD fimilar alfo. Therefore, &c.
d1.def. 6.
a 45. 1.
b 44. I.
€ 13.6.
d 18. 6.
e cor. 20. 6.
f 1.6.
g 14. 5.
a confir.
PRO P. XXV. Fig.62.
Unto the right-lined figure given ABEDC, to defcribe ano-
ther figure P, fimilar and alike fituate, which alfo fhall be
equal to another right-lined figure given F.
(a) Make the rectangle AL ÅBEDC; (b) alſo upon
BL make the rectangle BM= F; betwixt AB und BH
(c) find out a mean proportional NO; (e) upon NO (d)
make the polygon P fimilar to the right-lined figure
given ABEDC. I fay the polygon P fo made, fhall be
equal to F, that was given.
For ABEDC (AL): P::AB: BH:: (f) AL: BM.
Therefore P (g) —BM (h)—F. Which was to be
done.
PROP.
EUCLIDE's Elements.
91
PRO P. XXVI. Plate III. Fig. 60.
If from the parallelogram ABCD, be taken away ano-
ther parallelogram AGFE, fimilar to the whole, and alike
fituate, having also an augle EAG common with it;
then is that parallelogram about the fame diagonal AC
with the whole.
If you deny AC to be the common diagonal, then let
AHC be it, cutting EF in H, and let HI be drawn paral-
lel to AE. Then are the Pgrs. EI, DB, (a) fimilar; (b)
therefore AE: EH::AD: DC: :(c) AE: EF, and (d)
conſequently EHEF. (f) Which is abfurd.
PRO P. XXVII, Plate IV. Fig. 1.
Of all parallelograms AD, AG, applied to the fame
right-line A B, and wanting in figure by the prallelo-
grams CE, KI, fimilar and alike fituate to the Pgr. AD,
which is defcribed upon the half line, the greateſt is that
AD, which is applied to the half line being like to the
defect KI
For becauſe that GE (a) equal GC, if KI which is
common be added, (b) then is KE
KECI (c) = AM;
add CG which is common, (d) then is AG
Gnomon MBL ; (e) but the Gnomon MBLCE (AD)
Therefore AG AD. Which was to be demonftrated.
PRO P. XXVIII. Fig. 2.
a 24. 6.
b 1. def.6.
c hyp.
d 9. 5..
f
9. ax.
a 43. I.
b 2. ax.
to the
c 36. 1,
To a right-line given AB, to apply a parallelogram AP,
equal to a right-lined figure given C, deficient by a parallelo-
gram ZR, which is fimilar to another parallelogram given
Ì,* but it is neceſſary that the right-lined figure given C, to
which the Pgr. to be applied AP must be equal, be not grea-
ter than the Pgr. AF, which is applied to half the line, fince
the defects both of AF, which is apply'd to half the line, and
of AP the parallelogram to be applied, muſt be fimilar.
Bifect AB in E; upon EB (a) make the Pgr. EG like
to the Pgr. D; and (6) let EGC+I. (c) Make the
Pgr. NT I, and like to the Pgr. given D, or EG ; draw
the diameter FB; Make FO= KN, and FQ=KT;
thro' O and Q draw the parallels SR, QZ. Then is
the Pgr, AP that which was fought.
For
d 2.ax.
eg.ax.
27.6:
a 18.6.
b ſch. 45.1
C25.6.
92
The fixth Book of
•
d conft. &
24. 6.
e conflr.
f 3. ax.
g 2. ax.
h 43. 1.
a 18. 6.
b 25. 6.
ċ 3. I.
d confir.
e 24, 6.
f conftr.
g 3. ax,
h 36. 1.
k 43. 1.
1 2. and 1.
ax.
2 11. 2.
b 17.6.
a cor. 8. 6.
b cor. 20. 6.
€ 24..5.
d ſcb. 14.5.
€ 22. 6.
f 24.5.
gfch. 14. 5.
n 47. I.
For the Pgrs. D, EG, OQ, NT, ZR, are all (d) fimilar
one to the other, and the Pgr. EG ≈ (e) NT-| C:
C=(e)
OQ+C; (ƒ) wherefore Cto the Gnomon OBQ (g)
=AO+PG=(b) AO+EP=AP. Which was to be done.
PROP. XXIX. Plate IV. Fig 3.
Upon a right-line given A B, to apply a parallelogram
AN equal to a right-lined figure given C, exceeding by a
Pgr. OP, which shall be like to another Pgr. given D.
Bifect AB in E. Upon EB (c) make a Pgr. EG like to
the given one D, and (4) let the Pgr. HK-EG+C, and
like to the given one D, or to EG. Make FEL≈ (c)
IH; and (c) FGMIK. Thro' L, M, draw the paral-
lels MN and RN; and AR parallel to NM. Produce
ABP, GBO; draw the diameter FBN. Then is AN
the parallelograms required.
For the Pgrs. D, HK, LM, EG, are (d) fimilar; (e)
therefore the Pgr. OP is fimilar to the Pgr. LM, or D.
Alfo LM (f)=HK (ƒ) = EG+C. (g) Therefore C
to the Gnomon ENG. But AL (b) LB (k) =
AN. Which was to be done.
BM; (1) therefore C = AN.
PRO P. XXX. Fig. 5.
To cut a finite right-line given AB, according to extreme
and mean ratio (AB : AG :: AG : GB.)
(a) Cut AB in G, in fuch wife that ABXBG=AGq. (b)
Then BA: AG :: AG : GB. Which was to be done.
PROP. XXXI. Fig. 6.
In right-angled triangles BAC, any figure BF deſcribed
upon the fide B C, fubtending the right-angle BAC, is equal
to the figures BG, AL, which are fimilar and alike fituate
to the former BF, and defcribed upon the fides BA, AC,
containing the right-angle.
From the right-angle BAC let fall the perpendicular
AD. Becauſe DC: CA: : (a) CA: CB, therefore AL:
BF:: DC: CB. Alſo, becauſe DB: BA: :(a)BA:
BC, (b) therefore BG: BF: DB: BC; (c) therefore
AL BG: BF:: DC† DB (BC) : BC. (a) Therefore
ALBG= BF. Which was to be demonftrated.
Or thus: BG: BF :: (e) BAq : BCq: And (e) AL:
BF: ACq: BCq, (f) therefore B GAL: BF::
BAq+ACq: BCq. (g) Therefore whereas BAq+ACq
(b) BCq; (b) thence is BGAL-BF. Which was
to be demonftrated.
Coroll.
1
Plate III. facing Pag. 92.
[
Fig.
1.
E
G
Fig. 2. A
Fig. 3
A
F
G
F
B
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Fig. 4
I I
H
Fig. 5.
BH GAC G
F DK
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Fig. 7 G
FH
Fig. 8.
E
Fig. 6.
B
A
DE
B
D
K
Fig. 9. A G B
Z
B
B
M
L
H
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Fig. 10.A
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E
DH
C
D
A
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Fig. 13.
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C
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Fig. 14.
T
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B
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F
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Fig.51.
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K
Fig.
E
D
·CB
157
B
B
H
B F
JE
B
G
M
I
H
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BE
F
E Fig.58
H
K
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158.5
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F
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T
EUCLIDE's Elements.
93
Coroll.
From this propofition you may learn how to add
or fubtract, any like figures, by the fame method that
is uſed in adding and fubtracting of ſquares, in Schol.
47. I.
•
PROP. XXXII. Plate IV.
Fig. 4.
If two triangles ABC, DCE having two fides propor-
tional to two (AB: AC :: DC. DE) be fo compounded or
fet together at one angle ACD, that their homologous fides
are alſo parallel (AB to DC, and AC to DE,) then the re-
maining fides of thofe triangles (BC, CE) Jhall be found
placed in one frait-line.
:
For the angle A (a)= ACD) (a)= D, and AB : AC
(b):: DC : DE, (c) therefore the angle B=DCE,
Therefore the angle B-A (d) ACE; but the angle
B+A+ACB (e) 2 right, (f) therefore the angle AČE
ACB=2 right; (g) therefore BCE is a right-line.
Which was to be demonſtrated.
PRO P. XXXIII. Fig. 7, 8.
In equal circles DBCA, HFGP, the angles BDC, FHG,
bave the fame ratio with the peripheries BC, FG, on which
they ftand; whether the angles be fet at the centers (as BDC,
FHG) or at the circumferences, A, E: And fo likewife
bave the Sectors BDC, FHG.
Draw the right-lines BC, FG. Make CI=CB, and
GL=FG=LP, and join DI, HL, HP.
The arch BC (a)=CI, (a) alſo the arches FG, GL, LP,
are equal; (b) therefore the angle BDC=CDI, (b) and the
angle FHG=GHL=LHP. Therefore the arch BI is
the fame multiple of the arch EC, as the angle BDI is
of the angle BDC. And in like manner is the arch FP,
the fame multiple of the arch FG, as the angle FHP is of
the angle FHG. But if the arch BI —, —, — FP,
(c) then likewife is the angle BDI, FHP.
Therefore as the arch BC: FG: :(d) the angle BDC: FHG
BDC FGH
2
2
(e) :
Moreover, the angle BMC (g)=CNI; (b) and there-
fore the fegment B ČM CIN. (k) Alſo the triangle
BDC=CDI; (¹) wherefore the ſector BDCM=CDIN.
(ƒ):: A:E, Which was to be dem.
After
a 29. 1.
b bip.
c 6.6.
d 2. ax.
e 32. I.
f 1. ax.
g (4. I.
a 28. 3.
b 27.3.
c 27.3.
d6. def, 5.
e 15.5.
₤·20.3.
g 27. 3.
h 24. 3.
k 4. 1.
1 2. ax.
94
The fixth Book of
After the fame manner are the fectors FHG, GH L,
LHP equal one to the other. Therefore fince ac-
cordingly as the arch BI
the arch BI C, =, ~ F GP, fo is
FHP; (m) thence
m 6. def. 5. it will be as the ſector BDC: FHG:: the arch B´C: FG,
Which was to be demonftrated.
likewife the ſector BDI,
Coroll.
2 11. 5..
1. Hence, As fector (a) is to fector, fo is angle to angle.
2. The angle BDC in the center, is to four right-an-
gles, as the arch BC, on which it ftands, to the whole
circonference.
For as the angle BDC is to a right-angle, fo is the
arch BC to a quadrant. Therefore BDC is to four
right-angles, as the arch BC is to four quadrants, that
is, to the whole circumference. Alfo, as the angle A :
2 right: the arch BC: periphery.
:
3. Hence, the arches IL, BC, (Fig. 9.) of unequal circles
which fubtend equal angles, whether at the centers, as IAL
and BAC, or at the periphery, are like,
For I L periph. : : angle I A L (BAC): 4 right.
Alfo, Arch BC: periph.: :angle BAC : 4 right. There-
fore IL: periph.: BC: periph. And confequently
the arches IL and BC are fimilar. Whence
4. Two femidiameters AB, AC, cut off like arches I L,
BC from concentric peripheries.
The End of the fixth Book.
THE
[ 95 ]
The SEVENTH BOOK
O F
EUCLIDE's
I.
ELEMENT S.
Definitions.
Nity is that, by which every thing that is,
is called One.
U
II. Number is a multitude compoſed of
units.
III. Part is a number of a number; the leffer of the
greater, when the leffer meaſureth the greater.
Every part is denominated from the number, by which it
meaſures the number whereof it is a part; as 4 is called the
third part of 12, becauſe it measures 12 by 3.
IV. But when the leffer number does not meaſure
the greater, then the leffer is call'd, not a part, but
parts of the greater.
All parts whatſoever are denominated from theſe two num-
bers, by which the greatest common meaſure of the two num-
bers meaſures each of them; as 10 is faid to be two thirds of
the number 15; because the greatest common measure, which
is 5, meaſures 10 by 2, and 15 by 3.
V, A multiple is a greater number compared with a
leffer, when the leffer meaſures the greater.
VI. An even number is that which may be divided
into two equal parts.
VII. But an odd number is that which cannot be di-
vided into two equal parts; or that which differeth from
an even number by unity.
VIII. A number evenly even, is that which an even
number meaſureth by an even number.
IX. But a number evenly odd, is that which an even
number meaſureth by an odd number.
X. A number only oddly odd, is that which an odd
number meaſureth by an odd number.
XI.
96
The feventh Book of
XI. A prime (or firft) number is that which is mea-
fured only by unity.
XII. Numbers prime the one to the other, are fuch
as only unity doth ineafure, being their common meaſure.
XIII. A compofed number is that which fome certain
number meaſureth.
XIV. Numbers compofed the one to the other, are
thofe, which fome number, being a common meaſure
to them both, doth meaſure.
In this, and the preceding definition, unity is not a num-
ber.
XV. One number is faid to multiply another when
the number multiplied is fo often added to it ſelf, as
there are units in the number multiplying, and another
number is produced,
Hence in every multiplication unity is to the multiplier,
as the multiplicand is to to the product.
Obf. That many times, when any numbers are to be
multiplied (as A into B) the conjunction of the letters denotes
the product: So A BAX B, and CDE = CxD
× E.
XVI. When two numbers multiplying themfelves
produce another, the number produced is called a
plane number; and the numbers which multipled one
another, are called the fides of it: So 2 (C) × 3 (D) = 6
CD is a plane number.
XVII. But when three numbers multiplying one
another produce any number, the number produced is
termed a folid number; and the numbers multiplying
one another, are called the fides thereof: So 2 (C) X
3 (D) × 5 (E) = 30
=30=
CDE is a folid number.
XVIII. A ſquare number is that which is equally
equal; or, which is contained under two equal numbers.
Let A be the fide of a fquare; the Square is thus noted,
AA, or Aq.
XIX. A Cube is that number which is equally equal
equally; or which is contained under three equal num-
bers. Let A be the fide of a Cube; the Cube is thus no-
ted, AAA, or Ac.
In this definition, and the three foregoing, unity is num-
ber.
XX. Numbers are proportional, when the first is
the fame multiple of the fecond, as the third is of the
fourth; or, the fame part; or, when a part of the firſt
number meaſures the fecond, and the fame part of the
third
EUCLIDE's Elements.
97
third meaſures the fourth, equally: and vice versa. So
A: B::C: D. that is, 39
9: : 5:15.
XXI. Like plane, and folid numbers, are thoſe which
have their fides proportional: Namely, not all the fides,
but fome.
XXII.
A perfect Number is that which is equal to
its own parts.
As 6, and 28. But a number that is less than it's parts
is called an Abounding number, and one which is great-
er, a Diminutive: fo 12 is an abounding, 15 a diminutive
number.
XXIII. One number is faid to meaſure another, by a
third number, which when it either multiplies, or is
multiplied by the meaſuring number, produces the num-
ber meaſured.
In Divifion, unity is to the quotient, as the divifor is to
the dividend. Note, that a number placed under another
A
with a line between them, fignifies divifion: Só—— A di-
B
vided by B, and
CA
B
CXA divided by B.
Thoſe two numbers are called the Terms or Roots of
a Proportion, than which leffer cannot be found in the
fame proportion.
i.
T
Poftulates, or Petitions.
Hat numbers equal or multiple to any number
may be taken at pleaſure.
2. That a number greater than any other whatſoever
may be taken.
3. That Addition, Subtraction, Multiplication, Di-
vifion, and the Extractions of Roots or fides of fquare
and cube numbers, be alfo granted as poffible.
1.
W
Axioms.
Hatfoever agrees with one of many equal
numbers, agrees likewife with the reit.
2. Thofe parts that are the fame to the fame part, or
parts, are the fame among themſelves.
3. Numbers that are the fame parts of equal num-
bers, or of the fame number, are equal among them-
felves.
G
4. Thoſe
98
The feventh Book of
a 11. ax. 7.
b 12.ax. 7.
cg.ax. I.
4. Thofe numbers of which the fame number or
equal numbers, are the fame parts, are equal amongſt
themſelves.
5. Unity meaſures every number by the units that
are in it, that is, by the fame number.
6. Every number meaſures it fſelf by unity.
7. If one number multiplying another, produces a
third, the multiplier fhall meature the product by the
multiplied; and the multiplied fhall meaſure the fame
by the multipler.
Hence, No prime number is either a plane, folid, Square,
or cube number.
8. If one number meaſures another, that number by
which it meaſureth ſhall meaſure the fame by the units
that are in the number meaſuring, that is, by the num-
ber it felf that meaſures.
حرية
9. If a number meaſuring another, multiply that by
which it meafureth, or be multiplied by it, it produ-
ceth the number which it meaſureth.
10 How many numbers foever any number meaſureth,
it likewiſe meaſureth the numbers compofed of them.
11. If a number meaſures any number, it alſo mea-
fureth every number which the faid number meaſureth.
12. A number that meaſures the whole and a part ta-
ken away, doth alfo meaſure the refidue.
PROP. I.
A..... E. . G. BS 5 3
Ι
2
C...F.. D. ૐ I
H---
Tavo unequal numbers AB,
CD, being given, if the leſſer
CD, be continually taken from
the greater AB (and the refidue
EB from CD, &c.) by an alternate fubtraction, and the
number remaining never measure: the precedent, till unity
GB be taken; then are the numbers which were given AB,
CD, prime the one to the other.
If you deny it, let AB, CD, have a common meaſure,
namely the number H; therefore H meaſuring CD, doth
(a) alío meaſure AE; and (b) confequently the remainder
EB; (a) therefore it likewife meaſures CF, and (b) fo the
remainder FD; (a) therefore it alſo meaſures EG. But
it meaſured the whole EB, and (b) therefore it muſt mea-
fure that which remaineth GB, that is, a number mea-
fures unity. () Which is abfurd.
PROP.
EUCLIDE's Elements.
99
PROP. II.
Two numbers AB,
CD being given, not
prime the one to the other,
to find out their greateſt
common meaſure FD.
9
6
A
E………….. B 159 6
6
3
C......F
Ꭰ
2 3 1/3
G-..
Take the leffer number CD from the greater AB as
often as you can. If nothing remains, (a) it is manifeft
that CD is the greateſt common meaſure. But if there
remains fomething (as EB) then take it out of CD, and the
refidue FD out of EB, and fo forward till fome num-
ber (FD) meaſure the faid EB, (b) for this will be, be-
fore you come to unity; FD fhall be the greateſt common
meaſure.
For FD (c) meaſures EB, and (d) therefore alfo CF; and
(e) confequently the whole CD; (d) therefore likewiſe
AE; and fo meaſures the whole AB. Wherefore it is e-
vident that FD is a common meafure. If you deny it to
be the greateſt, let there be a greater (G); then whereas
G meaſureth CD, it (d) muſt likewife meaſure AE, (e)
and the refidue EB, (d) as alfo CF, (e) and by confequence
the refidue FD, (g) the greaterthe leſs. (b) Which is abfurd.
Coroll.
Hence, a number that meaſures two numbers, does
alſo meaſure their greateſt common meaſure.
PROP. III.
Three numbers being given, A, B,
C, not prime one to the other, to find out
their greatest common meaſure E.
B
A..
12.
8
D.... 4•
C ......6
E..2
If D does not mea-
Find out D, the greateſt common
meaſure of the two numbers A, B.
If D meaſures C the third, it is F...
clear that D is the greatest common
meaſure of all the three numbers.
fure C, at leaft D and C will be compoſed the one to
the other, by the Coroll. of the Propofition preceding.
Therefore let E be the greatest common meaſure of the
faid numbers D and C, and it ſhall be the number which
was required.
G 2
For
a 6.ax. 7.
b 1.7.
c cenftr.
d11.ax. 7.
e 12.ax. 7.
d 11.ax. 7.
d11.
e 12.ax. 7.
g Juppos.
h 9. ax. 1.
100
The Seventh Book of
•
a conft.
For E (a) meaſures C and D, and D meaſures A and B ;
b 11. ax. 7. therefore (b) E meaſures each of the numbers A, B, C:
c cor. I 7.
d ſuppos.
e 9. ax. 1.
a 4. def. 7.
b
3. def. 7.
€ 4. def. 7.
neither fhall any greater (F) meafure them; for if you
affirm that, (c) then F meaſuring A and B, does likewife
meaſure D their greatest common meafure; and in like
manner, F meaſuring D and C, does alfo meaſure E (c)
their greateſt common meafure, (d) the greater the lefs.
(e) Which is abfurd.
Coroll.
Hence, a number that meafures three numbers, does
alſo meaſure their greateſt common meaſure.
B.
B
• •
7
B...
18
PROP. IV.
Every less number A is of every
greater B either a part or parts.
If A and B be prime to one ano-
ther, (a) A fhall be as many parts of
the number B, as there are units
in A (as 6 of 7). But if A meaſures B, it is (4) plain
that A is a part of B (as 6 of 18.) Laftly, if A and B
be otherwife compofed to one another, (c) the greateſt
common meaſure fhall determine how many parts A
does contain of B; as 6 of 9.
.6
A......6
6
PROP. V.
4
D....4
4
a hyp.
b conft. &
ax. 1.
B......G...... C 12 E.... H.... F8.
If a number A be a part of a number BC, and another
number the fame part of another number EF; then both
the numbers together (A+D) shall be the fame part of both
the numbers together (BC+EF,) which one number Ă is of
one number BC.
For if BC be refolved into its parts BG, GC, equal
to A; and EF alfo into its parts EH, HF, equal to D;
(a) the number of parts in BC fhall be equal to the num-
ber of parts in EF. Therefore fince A+D (6) BG÷
EH-GC+HF, thence A+D fhall be as often in BC-j-
EF, as A is in BC. Which was to be demonftrated.
Or
EUCLID E's Elements.
101
X
Or thus. Let a=—,
2 b
x+y
y,
2
2
y
Ex, and c 2, ax, 1.
and b——, then (c) 2 a
then (c) 2 ax, and
2
therefore 2 a-|-2 bx+y, therefore a + b =
Which was to be demonſtrated.
PRO P. VI.
If a number AB be
parts of a number C,
and another number DE
the fame parts of ano-
3 3
A...G...B6 D.
4 4
A...G...B6 D. . . .H…... E 8
C.
C.... 9 F...
•
•
12
ther number F; then both numbers together AB+DE Ball
be of both numbers together C+F the fame parts, that one
number AB is of one number C.
a hyp.
b 5.7.
Divide AB into its parts AG, GB; and DE into its
parts DH, HE. The number of parts in both AB, DE,
is equal by fuppofition; fince then AG (a) is the fame part
of the number C, that DH is of the number F, AG +
DH (b) fhall be the fame part of the compounded num-
ber CF, that one number AG is of one number C.
(b) In like manner GB+HE is the fame part of the ſaid
CF, that one number GB is of one number C. (c) c2.ax. 7.
Therefore ABDE is the fame parts of C+F, that
AB is of C, Which was to be demonftrated.
2
3
Or thus. Let a =3x, and b
-2x, and b= y, and x-|- y=g,
then, becauſe 3 a 2 x, and 3 b= 2y, is 3 a +3b=
2x+2y2g, therefore a + b}g=}x+y•
PROP. VII.
If a number AB be
the fame part of a
number CD, that a
part taken away A E
is of a part taken away
6
5
3
A……... E . . . B. 8
ΙΟ
6
G.................F…………..D 16
CF; then ſhall the refidue EB be the fame part of the refi-
due FD that the whole AB is of the whole CD.
(a) Let EB be the fame part of the number GC that
AB is of CD, or AE of CF, (b) therefore AE+EB is
the fame part of CFGC that AE is of CF, or AB
of CD; (c) therefore GF CD. Take away CF com-
mon to both, and (d) there remains GC—FD. (e) Where-
G 3
fore
a i post. 7.
b 5.7.
c 6. ax. 1.
d
3. ax. 1.
e 2.ax. 7°
ΤΟΣ
The Seventh Book of
f 1. 2.
g bjp.
a 3. ax..
b conftr.
C 3. ax. 1.
27.7.
e 9. ax, 7.
f 1. 2.
g I. ax. 1.
ĥ byp.
k 3. ax. 1.
18. ax. 7.
fore EB is the fame part of the refidue FD (GC) that the
whole AB is of the whole CD. Which was to be dem.
C
Or thus. Let a bx; and cd=y; and x 3 Y,
in like manner as a=3c; I fay b 3 d. (f) For 3 c-+-
3 d
3y=x(g)=a+b, take away from both 3 cg)
a, and there remains 3 db. Which was to be dem.
PROP. VIII.
2
6 2 4 2
A...... H..G....E.. L.. B 16
6
18
C ...
F......D 24
If in a number AB
there be the fame parts of
a number CD, that a
part taken away AE, is
of a part taken aray
CF; the refidue alſo EB ſhall be the fame parts of the refidue
FD, that the whole AB is of the whole CD.
Divide AB into AG, GB, parts of the number CD ;
alfo AE into AH, HE, parts of the number CF; and
take GL-AH-HE, (a) wherefore HG EL. And
becauſe (6) AG-GB, (c) therefore HG LB. Now
whereas the whole AG is the fame part of the whole
CD that the part taken away AH is of the part taken
away CF, (d) the refidue HG or EL fhall be the fame
part alfo of the refidue FD that AG is of CD. In like
manner, becaufe GB is the fame part of the whole CD,
that HE or GL are of CF, (d) therefore the refidue LB
ſhall be the fame part of the reſidue FD that GB is of
the whole CD. Therefore EL+LB (EB) is the fame
parts of the reſidue FD, that the whole A B is of the
whole CD. Which was to be demonſtrated.
3 Y
2a.
Or thus more eaſily. Let a +bx, and c +dy.
Alſo y=x as well as c — a; or, (e) which is the fame
2 x; and 3 c
I fay db. For 3 c+
3d (f)=3y=2x (f)=2 a2 b. (g) Therefore 3 c+3d
=2a-f2 b; take away from each 3 c (b) =2a, and (k)
there remains 3 d=2
2 b; (4) therefore db. Which
was to be demonftrated.
A.... 4
4 4
B....G....C 8
PROP. IX.
If a number A be a part of a num-
ber BC, and another number D the
fame part of another number EF;
then alternately what part or parts
the firft A is of the third D, the fame
E……….., H.,.., F 10 part or parts shall the fecond BC be of
5
D..... 5
5
the fourth EF.
A
}
EUCLIDE's Elements.
103
A is fuppofed D, therefore let BG, GC, and EH,
HF, parts of the numbers BC, EF be equal; BG and
GC to A; and EH, HF to D. The multitude of parts
is put equal in both. But it is clear that BG is (a) the
fame part or parts of EH, that GC is of HF; (b) where-
fore EC (BGGC) is the fame part or parts of EF
(EHHF) that BG alone (A) is of EH alone (D.)
Which was to be demonftrated.
b
Or thus. Let a==
3 cd, then
d
a 1. ax. 7.
& 4.7.
b 5 or 6.7.
,
and c =
; or 3 ab, and
3
3
*
C*
3C
d
* 15.5.
a
за
b.
1
11
PROP. X.
A..G..B
C......6
5
4
5
D... . . H ..... E 10
F…….
•
15
If a number AB be parts of
a number C, and another num-
ber DE the fame parts of anc-
ther number F, then alter-
nately, what parts or part the
first AB is of the third DE,
the fame parts or part jhall the fecond C be of the fourth F.
AB is taken DE, and CF. Let AG, GB,
and DH, HE, be parts of the numbers C and F, viz. as
many in AB as in DE. It is manifeft that AG is the
fame part of C, that DH is of F; (a) whence alternately
AG is of DH, and likewiſe GB of HE, and (b) fo con-
jointly AB of DE the fame part, or parts, that C is of F.
Which was to be demonftrated.
Or thus. Let a=-
2b
2d
and c
; or 3a=2b,
3
3
C
3C
2d d
and 3 c=2d. Then is
11
a
за
2b b.
a 9.7.
b59.7.
PROP. XI.
If a part taken away AE be
to a part taken away CF, as
the whole AB is to the whole
CD, the refidue alfo EB fhall be
to the refidue F D, as the whole
AB is to the whole CD,
G4
4
3
A....E,.,B 7
8
6
C........F,...., D 14.
First,
104
The ſeventh Book of
a 4. 7.
b 20. def 7.
€7, or 8.7,
First, let AB be CD; (a) then AB is either a part
or parts of the number CD; and likewife AE is (b) the
fame part or parts of CF; (c) therefore the refidue EB
is the fame part or parts of the refidue FD that the
whole AB is of the whole CD, (b) and ſo AB : CD: :EB
: FB. But if AB be CD, then according to what is
already fhewn, will CD: AB: :FD: EB, therefore by
inverſion AB: CD::FB:FD. W.W'.D,
PROP. XII,
A, 4. C, 2.
B, 8. D, 4.
C, 2. E, 3.
F, 6.
the
antecedents A is to one of
tecedents (A+C+E) be to
If there be numbers, how many
foever, proportional (A : B : : C:
D:: EF;) then as one of the
confequents B, fo fhall all the an-
all the confequents (E+D+F.)
First, let A, C, E, be B, D, F; thon, becauſe
of the fame proportions, (a) ſhall A be the fame part or
parts of B that Cis of D; (b) and likewiſe conjointly A+
C fhall be the fame part or parts of B+D, that A alone
is of B alone. In the like manner A-CE is the fame
c 20. def. 7, part or parts of B+D+F that A is of B. (c) Therefore
ATCTE: B+D+F: A: B. But if A, C, E, be put
greater than B, D, F, the ſame thing may be fhewn by
inverfion. WW.D.
a 20. def. 7.
b 5. & 6, 7,
a 20. def. 7.
b 9.& 10.
A, 3. C, 4.
B, 9. D, 12.
PRO P. XIII.
If there be four numbers proportional
(ABCD) then alternately they
jhall also be proportional, (A : C:: B
: D.)
Firſt, let A and C be B and D, and A — C. By
reaſon of the fame proportion (a) A fhall be the fame
part or parts of B, that C is of D. (b) Therefore alter-
nately A is the fame part or parts of C, that B is of D;
and fo A: C:: B: D. But if A be C, and A and C
—
fuppofed Band D, it will come to the fame thing
by inverting the proportions. WW.D.
A, 9, D, 6.
B, 6. E, 4.
C, 3: F,
2.
PRO P. XIV.
If there be numbers, how many foever,
A, B, C, and as many more equal to
them in number, which may he compared
tavo and two in the ſame proportion (A : B
: D.
EUCLIDE's Elements.
105
:: D: E and B: C: E:F) they shall alfo by equality,
be in the fame proportion (A:C:
: D:F)
For becauſe A: B :: D: E, (a) therefore alternately is a 13. 7.
A:D: BE:: (a) C: F; (a) therefore again, by per-
mutation, A:C: :D : F.
ftrated.
Which, was to be demon-
PROP. XV.
If an
an unite meaſure any number
B, and another number D equally
meafure fome other number E; alter-
I.
D..z
B... 3. E......6.
nately alfo fhall an unite meaſure the third number D, as
often as the fecond В doth the fourth E.
For feeing is the fame part of B, that D is of E;
(a) therefore alternately fhall be the fame part of D, a 9.7.
that R is of E. Which was to be demonftrated.
PRO P. XVI.
If two numbers A, B, mutually
multiplying themſelves, produce any
numbers AB, BA; the numbers pro-
duced AB, and BA, ſhall be equal
the one to the other.
A, 3.
B, 4.
B, 4-
A, 3.
AB, 12. BA, 12.
be as
a 15.def.7.
b 15.7.
For becauſe AB = A x B, (a) therefore ſhall
often in A, as B in A B, (b) and by confequence alter-
nately fhall be as often in E as A in A B. But becauſe
BA=B x A, (a) therefore fhall be as often in B, as
A in BA, therefore as often as 1 is in AB, fo often is I
in BA, and (c) ſo AB=BA. Which was to be demon-
firated.
PRO P. XVII.
A, 3-
B, 2.
AB, 6.
C, 4.
AC, 12.
Ifa number A multiplying two
numbers B,C, produce other num-
bers AB, AC; the numbers pro-
duced of them shall be in the fame
proportion that the numbers multiplied are. (AB : AC: : B:C.)
For fince AB-Ax B (a) therefore fhall be as often
in A as B in AB. (a) Likewiſe becauſe AC= AXC
fhall be as often in A, as C in AC, and fo alfo
B as often in AB as C in AC ; (b) wherefore B: AB::
C: AC, (c) and therefore alſo alternately B: C::AB:
AC. Which was to be demonſtrated.
€ 4. ax. 7.
a 15. def.7
b 20.def.7.
c 13. 7-
ROP.
гоб
The feventh Book of
t
a 16. 7.
b 17.7.
C, 5.
A, 3.
C, 5.
B, 9.
PROP. XVIII.
If two numbers AB, multiplying
any number C, produce other num-
bers AC, BC; the numbers pro-
d:ced of them shall be in the fame
proportion that the numbers multiplied are. (A: B:: AC:
AC, 15. BC, 45.
BC.)
For (a) AC CA, and BC (a) CB; fo the fame C
multiplying A and B produceth AC and BC ; (b) there-
fore ABAC : BC. Which was to be demonftrated.
a 17. 7.
b hyp.
€ 18. 7.
d 9.5.
e byp.
f 17.5.
g 17.7.
h 18.7.
K 11. 5.
Schol.
Hence is deduced the vulgar manner of reducing
fractions (,,) to the fame denomination. For mul-
tiply 9 both by 3 and 5, and they produce 223;
becaufe by this 3:5:27:45. Likewife multiply 5 by
7 and 9, there arifes &; becaufe 7:9:35:45.
3 S
5
PROP. XIX.
A, 4. B, 6. C, 8
D,
12,
If there be four numbers
AD, 48. BC, 48. in proportion (A: B::C:
D.) the rectangle or number
produced of the first and fourth (AD) is equal to the rectangle
or number which is produced of the fecond and third (BC.)
And if the number which is produced of the first and fourth
(AD) be equal to that produced of the fecond and third (BC)
thofe four numbers fhall be in proportion (A: B: C: D.)
1. Hyp. For AC AD (a) :: C:D (b) :: A: B(c)
:: AC: BC; (d) therefore AD = BC.´ Which was to
be demonftrated.
:
2. Hyp. Becaufe (e) ADBC, therefore AC: AD (f)
: AC BC. But AC: AD(g): : C: D, and AC: BC
(b) A: B; (k) therefore CD: : A: B. Which
was to be dewonftrated.
X
PROP.
EUCLIDE's Elements.
107
PRO P. XX.
A.
4-.
B.
C.
6. 9.
AC, 36. BB, 36.
D, 6.
If there are three numbers in pro-
portion (A : B :: B: C) the number
contained under the extremes (A C)
is equal to the Square made of the
middle (BB). And if the number
contained under the extremes be eqnal to that (Bq,) pro-
duced of the middle, thofe three numbers fall be in proportion
A: B B: C.
1. Hyp. For take DB, (a) therefore A: B: :D (B)
: C (b) wherefore ACBD, (a) or BB. Which was to
be demonftrated.
2. Hyp. Becauſe AC (c) = BD, (b) therefore A: B: :
D (B): C. Which was to be demonftrated.
PROP. XXI.
Numbers A B, C D,
E....
A...G..B5.
C..H.D 3. F......6.
10.
being the least of all that
have the fame proportion
with them (E, F,) equally meaſure the numbers E, F,
having the fame proportion with them; the greater AB the
greater E, and the leffer CD, the leſſer F.
For AB: CD (a): : E:F;(b) therefore alternately AB :
E:: CD: F; (c) therefore AB is the fame part or parts
of E that CD is of F; but parts it cannot be, for if fo,
then let AG, GB, be parts of the number E; and CH,
HD, parts of the number F, (c) therefore AG : E:
CH: F, and by inverfion AG: CH (d) :: E: F, (e)
::AB: CD; therefore AB, CD, are not the leaft in
their proportion; which is contrary to the hypothefis.
Therefore, &'c:
PROP. XXII,
If there are three numbers A, B, C ;
and other numbers equal to them in num-
ber, D, E, F; which may be compa-
red two and two in the fame propor-
(
A, 4. D, 12.
B, 3. E, 8.
C, 2. F, 6.
tion: and if alſo the proportion of them be perturbed (A:B::
E: F, and B:C:: D: E.) then by equality they shall be
in the fame proportion (A: C: :D: F.)
For
a 1. a*. 7.
b 19. 7.
c hyp.
d 19.7.
a hyp.
b 13. 7.
c 20. aef.
d 13. 7.
e hyp.
108
The ſeventh Book of
a hyp.
b 19.7.
C} ax. I.
d 19. 7.
2 21.7.
For becauſe A B (a): E: F, therefore fhall AF
:
EF, =
BE; and becauſe B: C:: (a) D: E, (b) therefore BE=
CD, (c) and confequently AF CD. (d Therefore
A: C: :D:F. Which was to be demonfirated.
A, 9. B, 4.
C -- D --
---
E-
PROP. XXIII.
Numbers prime the one to the other,-
A, B, are the least of all numbers
that have the fame proportion with
them.
If it be poffible, let C and D be leſs than A and B,
and in the fame proportion; (a) therefore C meaſures A
equally as D meafures B, fuppofe by the fame number E;
b 23. def. 7. and fo C fhall be (b) as often in A as I is in E, (c) like-
CI5.7.
a 9. ax. 7.
b 17.7.
wife alternately, E as often in A as I in C. By the
like reaſoning as many times as 1 is in D, fo many times
fhall E be in B. Therefore E meaſures both A and B ;
which confequently are not prime the one to the other,
contrary to the hypothefis.
A, 9. B, 4.
C---
D---E-
PRO P. XXIV.
Numbers A, B, being the leaft of
all that have the fame proportion with
them, are prime the one to the other.
If it be poffible, let A and B have a
common meaſure C; and let the fame meaſure A by D,
and B by E; (a) therefore CD=A, (b) and CE=B. (b)
Wherefore A:B::D: E. But D and E are leffer than
A and B, as being but parts of them. Therefore A
and B are not the leaft in their proportion, againſt the
Hypothefis.
A, 9. B, 4.
C, 3. D--
number B.
PROP. XXV.
If two numbers A, B, are prime the
one to the other, the number Cmeaſuring
one of them A, fhall be prime to the
For if you affirm any other D to meaſure the numbers
a 11.ax. 7. B and C, (a) then D meaſuring C does alfo meaſure A,
and confequently A and B are not prime the one to the
other. Which is against the Hypothefis.
PROP.
EUCLIDE's Elements.
109
PROP.
XXVI,
A, 5.
C, 8.
B, 3.
AB, 15.
E-
F.
If two numbers A, B, are prime
to any number C, the number alfo
produced of them AB, ſhall be prime
to the fame C.
If it be poffible, let the number E be a common meaſure
AB
E
to A B, and C; and let be = F; (a) thence AB
EF; (6) wherefore alfo F: A:: B: F, But becauſe A
is prime to C, which E meaſures, (c) therefore E and A
are prime to one another, (d) and fo leaſt in their own
proportion, (e) and confequently they muft meafure B and
F; namely F fhall meaſure B, and A fhall meaſure F.
Therefore feeing E meaſures both B and C, they ſhall
not be prime to one another: Contrary to the Hypothefis.
PROP. XXVII.
If two numbers A, B, are prime to
one another, that alfo which is produced
of one of them (Aq) ſhall be prime to the
other B.
A, 4.
Aq, 16.
D, 4·
B, 5·
Take DA; therefore both D, and A are prime to
B; (6) therefore AD or Aq is prime to B. Which was
to be demonftrated.
PROP. XXVIII.
If two numbers A, B, are prime
to two numbers C, D, each to either
of both, the numbers alſo produced by
multiplying them AB, CD, ſhall be
prime to one another.
A, 5.
C, 4.
D, 2.
B, 3:
AB, 15. CD, 8.
For becauſe A and B are prime to C, (a) therefore
fhall AB alſo be prime to the fame. And for the fame
reaſon ſhall AB be prime to D. (b) Therefore AB is
prime to CD. Which was to be demonftrated.
PROP. XXIX.
If two numbers A, B, are prime
to one another, and each multiply-
ing itself produces another number.
(Aq, and Bq;) then the numbers pro-
A, 3-
B, 2.
Aq; 9.
Bq, 4.
Ac, 27.
Bc, 8.
duced
a 9. ax. 7.
b 19. 7,
C25. 7,
d 23. 7.
e 21.7.
a 1. ax. 7.
b 26. 7.
a 26. 7.
b 26. 7.
J10
The feventh Book of
a 27. 7.
b 28.7.
duced of them (Aq, Bq) fhall be prime to one another. And
if the numbers given at first A, B, multiplying the faid pro-
duced numbers (Aq, Bq,) produce others (Ac, Bc,) those
numbers alſo ſhall be prime to one another: And jo on.
For becauſe A is prime to B, (a) therefore Aq fhall
be prime to B, and Aq being prime to B, (a) therefore
Aq fhall be alfo prime to Bq. Again, becauſe A is as well
prime to B and Bq, as Aq is to the faid B and Bq, (b)
therefore fhall AxÀq, that is, Ac, be prime to BxBq,
that is, to Bc: And fo forth of the reſt. W.W.D.
8
5
PROP. XXX.
A ....... B . . . . . C 13. D -
If tavo numbers AB,
BC, be prime the one to
the other; then both ad-
either of them AB, BC.
ded together (AC) ſhall be prime to
And if both added together AC be prime to any one of them
AB, the numbers alſo given in the beginning AB, BC, ſhall
be prime to one another.
i. Hyp. For if you would have AC, AB to be com-
a 12.ax. 7. pofed, let D be the common meaſure: (a) this fhall mea-
fure the refidue BC: And therefore AB, BC, are not
prime to one another; which is against the Hypothefis.
2. Hyp. AC, AB, being taken prime to one another,
b 10. ax. 7. let D be the common meaſure of AB, BC. (b) But feeing
that meaſures the whole AC, therefore AC, AB, are not
prime to one another; contrary to the Hypothefis.
Coroll.
Hence, a number, which being compounded of two,
is prime to one of them, is alfo prime to the other.
A, 5. B, 8.
PRO P. XXXI.
Every prime number A is prime to every
number B, which it meaſureth not.
For if any common meafure doth meaſure both, A.
2 11. def, 7. B, (a) then A will not be a prime number; contrary to the
Hypothefis.
A, 4. D, 3.
B, 6. E, 8.
AB, 24.
PROP. XXXII.
If two numbers A, B, multiplying one
another produce another AB, and fome
prime Number D, measure the number
produced of them AB; then ſhall it alſe
meaſure one of thoſe numbers, A, or B, which were given at
the beginning.
Suppoſe
{
EUCLID E's Elements.
III
Suppofe the number D not to meaſure the number A,
AB
and let be E, (a) then AB≈DE; (b) whence D :
beE,
whence D:
D
A:: B: E. (c) But D is prime to A; (d) therefore D and
A are the leaft in their proportion; (e) and confequently
D meaſures B as often as A meaſures E. The propofition
therefore is evident.
PROP. XXXIII.
Every compofed number A, is measured by
fome prime number B.
A, 12.
B, 2.
Let one or more numbers (a) meafure A, of which let
the leaſt be B; that ſhall be a prime number: For if it
be faid to be compofed, then fome (a) leffer number
fhall meaſure it, (b) which fhall alfo confequently meaſure
A. Wherefore B is not the leaft of thofe which meaſure
▲, contrary to the Hyp.
PRO P. XXXIV.
Every number A, is either a prime, or meaſured
by fome prime number.
A, 9.
For A is neceffarily either a prime or a compofed
number. If be a prime, 'tis that we affirm. If com-
pofed, (a) then fome prime number meaſureth it. Which
was to be demonftrated
a 9. ax. 7.
b 19. 7.
c hyp. and
31.7.
d 23. 7.
e 21.7.
a 13. def. 7.
b11.ax.7.
a 33. 7.
PRO P. XXXV,
A, 6.
B, 4. C, S.
D, 2.
H--I--K---
E, 3. F, 2.
G, 4, L-.-
How many numbers foever A, B, C, being given to find
the least numbers E, F, G, that have the fame proportien
with them.
If A, B, C, be prime to one another, (a) they fhall
be the leaft in their proportion. If they be compofed,
(b) let their greateft common meaſure be D, which let
meaſure them by E, F, G. Theſe are then the leaft
in the proportion A, B, C.
For D x E, F, G, (c) produceth A, B, C, (d) therefore
thefe and thoſe are in the fame proportion But allow
other numbers H, I, K, to be the leaft in the fame pro-
portion
a 23 7.
b 3. 7.
cọ ax.7.
d 17. 7.
112
The Seventh Book of
e 21. 7.
£ 9. ax. 7.
gr. ax. 1-
portion; (e) which fhall therefore equally meaſure A, B,
C, namely by the number L, (f) therefore L x H, I, K,
fhall produce A, B, C, (g) and confequently ED = A=
HL; (b) from whence E: H::L:D.
L:D. But E(k)
But E(k)!
H; (1) therefore LD, and fo D is not the greateſt
I 20. def. 7. common meaſure of A, B, C. Which is against the Hy-
h 19. 7.
k suppos.
a 9. ax. 7.
& 1. ax. 1.
b 19.7.
c hyp.
d 23. 7•
e 21. 7.
£17. 7.
g 20. def. 7.
h 35.7.
k 19.7.
17. ax. 7.
m 9. ax. 7.
n 19. 7.
o conftr.
P 21 7.
q 17. 7.
x 20. def. 7.
pothefis.
Coroll.
Hence, The greateft common meaſure of how many
numbers foever, meaſures them by the numbers which
are leaſt of all that have the fame proportion with them.
Whence is derived the vulgar method of reducing fracti-
ons to their leaft terms.
PROP. XXXVI.
Two numbers being given, A, B, to find out the least
number which they measure.
A, 5. B, 4.
AB, 20.
D-
E---F-
1. Cafe. If A and B be prime the one
to the other, AB is the number requi-
red. For it is manifeft that A and B
meaſure AB. If it be poffible, let A
and B meafure fome other number D
AB, fuppofe by E, and F; (a) therefore AE=D=
BF, (b) and ſo A : B :: F: E.
F: E. But becauſe A and B (c)
are prime one to the other, (d) and fo leaſt in their pro-
portion, A ſhall (e) equally meaſure F as B does E. But
B: E(ƒ):: AB: AE (D.)(g) Therefore AB fhall alſo
meaſure D, which is lefs than it felf. Which is abfurd.
A, 6. B, 4, F----
C, 3. D, 2.
AD, 12.
G---H---
2. Cafe. But if A and
B be compofed one to a-
nother, (b) let there be
found C and D the leaſt
in the fame proportion. (k) Therefore AD BC; and
AD or B fhall be the number fought.
For it is (plain that B and A meaſure AD or BC.
Conceive A and B to meaſure FAD, namely A by
G, and B by H, (m) therefore AG F BH ; (n)
whence A: B: H:G(6)::C: D, (p) and confequent-
ly C equally meaſures Has D does G. But D: G (9)
::AD: AG(F,) therefore AD (r) meaſures F, the
greater the leſs. Which is abfurd.
Corell,
EUCLIDE's Elements.
113
Coroll.
Hence, If two numbers multiply the leaft that are in
the fame proportion, the greater the lefs, and the lefs the
greater, the leaft number which they meaſure fhall be
produced.
PROP. XXXVII.
If two numbers A, B, meaſure
any number CD, the least number
which they meaſure, E, ſhall alſo
meaſure the fame CD.
A, 2. B, 3.
E......6.
C-- --F - - - D.
If you deny it, take E from CD as often as you can,
and leave FDE, therefore ſeeing A and B (a) meaſure
E, (b) and E meaſures CF, (c) likewife A and B will mea-
fure CF. But (a) they meaſure the whole CD; (d) there-
fore alſo they meaſure the refidue FD; and confequently
E is not the leaſt which A and B meaſure: Contrary to the
Hypothefts.
PROP. XXXVIII.
Three numbers being given, A,
B, C, to find out the leaft which
they meaſure.
A, 3. B, 4. C, 6.
D, 12.
(a) Find D the leaft that two of them A and B mea-
fure; which if the third C alfo meaſure, it is manifeft
that D is the number fought. But if C doth not
meaſure D, let E be the leaſt that C and D meaſure,
E fhall be the number required.
C, 4.
A, 2. B, 3.
D, 6. E, 12.
F--
For it appears by the 11. ax.
7. that A, B, C, meaſure E;
and it is eaſily ſhewn that they
meaſure no other F lefs than E.
For if you affirm they do, (b) then D meaſures F, (b)
and confequently E meaſures the fame F, the greater the
lefs. Which is abfurd.
Coroll.
Hence it appears, that if three numbers meaſure any
number, the leaſt alſo, which they meaſure, ſhall
meafure the fame.
H
PROP.
a hyp.
b conftr.
c 11.ax.7.
d 12.ax-7.
a 36. 7.
b 37.7.
114
The feventh Book of
PROP. XXXIX.
A, 12.
If any number B, meaſures a number As
B, 4. C, 3. the number meafured A, Jhall have a part
C denominated of the number measuring B.
A
For becauſe — (a) = C, (b) therefore A = BC (c) and
a hyp.
b 9. ax. 7.
c7.ax. 7.
A
Ĉ
= B.
B
Which was to he demonftrated.
PROP. XL.
A, 15.
If a number A, have any part whatfoever
C, 5. B, the number C, from which the part
B is denominated, fhall meaſure the ſame.
B, 3.
A
B. Which
a hyp. & 9.
For fince BC (a) = A, (b) therefore
ax. 7.
b 7. ax. 7.
was to be demonſtrated.
PRO P. XLI.
a 38.7.
b 39.7.
0 40. 7.
HK-MAIL
I
G, 12.
H
1
To find out a number G, the least that can
bave given parts, ž›ŝi 4
I I I
·
(a) Let G be found the leaſt which the denominators
2, 3, 4, meaſure; (b) it is evident that G has the parts
4. If it be poffible let HG have the ſame
parts; (c) therefore 2, 3, 4, meaſure H; and fo G is
not the leaft which 2, 3, 4, meafure: against the fuppofition.
The End of the feventh Boo K.
The
[ 115 ]
The EIGHTH BOOK
O F
EUCLIDE's
ELEMENT S.
I'
PROP. I.
A, 8. B, 12, C, 18. D, 27.
E-F--G---H-
F there be divers numbers how many foever in continual
proportion, A, B, C, D, and their extremes A, D,
prime to one another; then those numbers A, B, C, D,
are the least of all numbers that have the fame propor-
tion with them.
For, if it be poffible, let there be as many others E,
F, G, H, less than A, B, C, D, and in the fame propor-
tion with them. (a) Therefore from equality A: D::
E: H, and fo A and D, which are prime numbers, (b)
and confequently the leaft in their proportion, (c) equal-
ly meaſure E and H, which are lefs than themſelves.
Which is abfurd.
Ac, 8.
PRO P. II.
I
A, 2. B, 3.
Aq, 4. AB, 6. Bq, 9.
AqB, 12. BqA, 18. Bc, 27.
To find out the least numbers continually proportional, as
many as ſhall be required, in the proportion given of A to B.
Let A and B be the leaft in the proportion given;
Then Aq, AB, Bq, fhall be the three leaft in the fame
continual proportion that A is to B.
For AA: AB (a) :: A : B ( b) :: AB: BB. Likewife
becauſe A and B are prime one to the other, (c) fhall Aq,
Bq, be alfo prime to one another, (d) and fo Aq, AB, Bq,
are the leaft in the proportion of A to B.
H 2
More-
à 14. 7.
b 23.7.
C 21.7°
a 17. Že
b 24.7.
c 29. 7.
d 1, &
116
The eighth Book of
€ 17.7
f 29.7.
g 1. 8.
Moreover, I fay Ac, AqB, BqA, BC, are the four leaft
in the proportion of A to B. For AqA: AqB (e): : A:
B(e): ABA (AqB.): ABB, (e) and A: B: ABqA :
BBB (Bc.) Therefore fince Ac, and Bc, are (ƒ) prime to
one another, likewife (g) fhall Ac, AqB, BqA, Bc be the
four leaft in the proportion of A to B. In the fame
manner may you find out as many proportional numbers
as you pleaſe. Which was to be done.
Coroll.
1. Hence, If three numbers, being the leaſt, are pro-
portional, their extremes fhall be fquares; if four, cubes.
2. The extremes of any number of proportionals
found by this propofition, if fuch proportionals are the
least of all in a given ratio, are prime to one another.
3. Two numbers, being the leaft in a given ratio,
meaſure all the mean numbers of proportionals be they
ever fo many, provided they are the leaft in the fame
proportion; becauſe they arife from the multiplication
of them into certain other numbers.
4. Hence it also appears by the conftruction, that the
ſeries of numbers 1, A, Aq, Ac.; 1, B, Bq, Bc; Ac, AqB,
BqA, Bc, confift of an equal multitude of numbers; and
confequently, the extreme numbers of how many foever
the leaft continually proportionals are the last of as many
other continually proportionals from unity; thus the ex-
treines Ac, Bc, of the continual proportionals Ac, AqB,
BqA, Bc, are the laft of as many proportionals from uni-
ty 1, A, Aq, Ac, and 1, B, Bq, Bc.
5. 1, A, Aq, Ac; and B, EA, AqB; and Bq, BqA are
in the ratio of 1 to A. Alfo B, Bq, Bc; and A, AB,
BqA; and Aq, AqB are in the ratio of 1 to B.
a 2. 8.
PROP. III.
A, 8. B, 18. C, 12. D, 27.
If there be numbers
continually proportional,
how many foever, A, B, C, D, being also the least of all
that have the ſame proportion with them; their extremes A,
D, are prime to one another.
For if there be (a) found as many numbers the leaft in
the proportion of A to B, they fhall be no other than
A, B, C, D; therefore, by the ſecond Coroll. of the pre-
cedent prop. the extremes A and D are prime to one ano-
ther. Which was to be demonſtrated.
PROP.
EUCLIDE's Elements.
117
Proportions bow
PROP. IV.
ma-
A, 6. B, 5. C, 4, D.3.
H, 4. F, 24. E, 20. G,15
I--K--L-
a 36.7.
ny foever being given in the
leaft numbers (A, to B,
and C to D) to find out the
leaft numbers continually proportional in the proportions given.
(a) Find out E, the leaft number which B and C mea-
fure; and let B meaſure E (b) as often as A does another
F, viz. by the fame number H. (b) Alfo let C meaſure the
faid E as often as D meaſures another G, then F, E, G,
fhall be the leaft in the proportions given. For AH (c) c 9. ax.
F, and BH (c) E; (d) therefore A: B:: AH : BÍ
E, 5.
F, 7-
pro-
b 3 post. 7.
d 18. 7›
e 7.5.
8 37.7.
(e) : : F: E. In like manner C:D: :E: G ; therefore
F, E, G, are continually proportional in the proportions
given. And they are moreover the leaft in the faid
portions; for conceive other numbers I, K, L, to be the
leaft ; (f) then A and B muft equally meaſure I and K, (f) f 21.7.
and C and D likewife K and L; and fo B and C mea-
fure the fame K. (g) Wherefore alfo E meaſures the fame
number K, which is lefs then it ſelf. Which is abfurd.
A, 6. B, 5. C, 4. D, 3.
H, 24. G, 20. I, 15.
But three proportions being given, A to B, C to D,
and E to F; find out as before three numbers H, G, I,
the leaſt continually in the proportions of A to B, and C
to D. Then if E meaſures I, (b) take another number K,
which may be equally meaſured by F; and thoſe four
numbers H, G, I, K, fhall be continually the leaſt in the
given proportions; which we need go no other way to
prove than we did in the first part.
K, 21.
A, 6. B, 5. C, 4. D, 3. E, 2.
H, 24. G, 20. I, 15.
N, 105.
M, 48. L, 40. K, 30.
F, 7•
If E doth not meaſure I, let K be the leaft which E and
I meaſure; and as often as I meaſures K, let G as often
meaſure L, and H alfo M, fo likewife let F meaſure N
often as
E meaſures K. The four numbers M, L, K,
N, ſhall be the leaft continually in the given proportions;
which we may demonftrate as before.
as
7.
h 3 post. 7.
PROP.
H 3
118
The eighth Book of
C, 4.
E, 3.
PROP. V.
D, 6. F, 16. ED, 18.
CD, 24. EF, 48.
Plang numbers CD, EF,
are in that proportion to one
another which is compofed
of their fides.
For becaufe CD: DE (a) : : C ; E; (a) and ED : EF
CD ED
EDEF
a 17. 7.
b20 def. 5 : :D :
CD
:: D: F and
c 11. 5.
EF (6)
LUXEF (c) then ſhall be the
CD
D
proportion
EF
F.
a 20. def. 7.
b 35.7.
'c 5.7.
d 3, 3.
€ 14. 7.
a 6.7
Which was to be demon.
PROP. VI.
A, 16. B, 24. C, 36. D, 54, E, 81.
F, 4. G, 6. H, g.
If there be num
ber's continually pro,
portional how many
foever, A, B, C, D, E, and the first A does not measure the
fecond B, neither fhall any of the other measure any one of
the rest.
Since A does not meaſure B, (a) neither fhall any one
meaſure that which next follows; Becaufe A: B:: B:C
: : C : D, &c. (b) Take three numbers, F, G, H, the leaſt
in the proportion of A to B, therefore fince A does not
meaſure B, (a) neither ſhall F meaſure G, (c), therefore
F is not unity. But F and H are prime one to another;
therefore (fince by equality AC:: F: H, and F
does not meaſure H, (a) neither fhall A meaſure C; and
confequently neither ſhall B meaſure D, nor C meaſure
E, &c. becauſe A: C (e) : : B : D (e) : : C : E, &c,
In like manner four or five numbers being taken the
leaft in the proportion of A to B, it may be fhewn that
A does not meafure D and E; nor does B meaſure E and
F, &c. Wherefore none of them ſhall meaſure any other.
Which was to be demonftrated.
PROP. VII.
A, 3. B, 6. C, 12. D, 24. E, 48,
If there be numbers continually proportional how many foe-
ver A, B, C, D, E, and the first A measures the last E, it
fhall also measure the fecond B.
If you deny that A meaſures B, (a) then neither ſhall
it meaſure E; Which is contrary to the Hyp,
PROP.
EUCLIDE's Elements.
119
If between two
numbers A,B, there
fall mean numbers
in continual propor-
PROP. VIII.
A, 24. C, 36.
G, 8. H, 12.
E, 32. L, 48.
D, 54. B, 81,
I, 18. K, 27.
M, 72. F, 108,
:
:
tion C, D; as many mean numbers in continual proportion
as fall between them, fo many mean numbers alſo L, M,
in continual proportion, ſhall fall between two other numbers
E, F, which have the fame proportion with them (L. M,)
(a) Take G, H, I, K, the leaft in the proportion of
A to C; (b) by equality it fhall be G: K: A B (c): :
E: F. But G, and K (d) are prime one to another. (e)
Wherefore G meaſures E as often as K does F. Let H
meaſure L, andI likewife M by the fame number; (f) there-
fore E, L, M, F, are in fuch proportion as G, H, I, K,
that is, as A, B, C, D. Which was to be demonſtrated.
PROP. IX.
If two numbers A,
В are prime to one ano-
ther, and mean num-
bers in continual pro-
1.
E, 2. F; 3-
G, 4. H,6. I, 9..
A, 8. C, 12. D, 18. B, 27.
portion C, D, fall between them; as many mean numbers
in continual proportion as fall between them, fo many means
alfo in continual proportion (E, G; and F, I) hall fall be-
tween either of them and unity.
It is evident, that 1, E, G, A, and 1, F, I, B, are,
and as many as A, C. D, B, namely by the 4th Coroll.
2.8. Which was to be demonftrated.
a 35.7.
b 14. 7.
c hyp.
d 3. 8.
e 21. 7.
£ conſtr
PROP.
X.
If between two nam-
bers A, B, and an unit,
numbers continually pro-
portional (E, D, and
F,G,) fall as many
A, 8. I, 12.
E, 4. DF, 6. G, 9.
K, 18. B, 27.
D, 2.
2. F, 3.
I.
mean numbers in continnal proportion as fall between either of
them and unity, ſo many means alſo ſhall fall in continuat
proportion between them, I, K.
For E, DF, G, and A, DqF (I) DG (K) B are →
by 2. 8, therefore, &c.
H 4
PROP.
120
The eighth Book of
PRO P. XI.
Aq, 4.
A, 2. B, 3.
AB, 6. Bq,9.
and Aq to Bq, is in duplicate
fide B.
Between two fquare num
bers Aq, Bq, there is one
mcan proportional number AB:
proportion of the fide A to the
a 17.7.
b 10. def. 5.
(a) It is manifeft that Aq, AB, Bq, are; (5) and
confequently alfo
A q
A
twice.
Which was to be
B q
B
1
demonftrated.
PROP. XII.
a 2. 8.
Ac, 27. AqB, 36. BqA,48. Bc, 64.
A, 3. B, 4.
Aq, 9. AB, 12.
Bq, 16.
bers AqB, BqA ; and the cube Ac is to
plicate ratio of the fide A to the fide B.
(a) For Ac, AqB. BqA, Bc, are
b 10. def. 5 of A to B ; (b) and therefore
A c
Between two cule
numbers, Ac, Bc,
there are two mean
proportional num-
the cube Bc in tri-
in the proportion
A
thrice. Which
B c
B
was to be demonfirated.
a 2.8.
b 14.7.
PROP. XIII.
A, 2. B, 4.
C, 8.
Aq, 4. AB, 8. Bq, 16. BC, 32. Cq, 64.
AC,8.AqB, 16.BqA,32. Bc,64, BqC,128. CqB,256. Cc, 512.
If there be numbers in continual proportion how many foever
A, B, C; and every of them multiplying itself produceth cer-
tain numbers; the numbers produced of them Aq, Bq, Cq,
Shall be proportional: And if the numbers firft given A, B, C,
multipling their products Aq, Bq, Cq, produce other numbers
Ac, Bc, Cc, they alſo fhall be proportional; and this fhall
ever happen to the extremes.
For Aq, AB, Bq, BC, Cq (a) are ; (b) therefore by
equality Aq: Bq: : Bq : Cq. Which was to be demonft.
(a) Alfo Ac, AqB, EqA, Bc, BqC, CqB, Cc, are ÷ ; (b)
therefore again by equality Ac Bc: Bc: Cc. Which
was to be demonftrated.
PROP.
EUCLIDE's Elements.
121
PROP. XIV.
If a Square number Aq mea-
fure a fquare number Bq, the fide
alfo of the one (A) ſhall meaſure
:
Aq. 4. AB, 12. Bq,36.
A, 2.
B, 6.
the fide of the other (B) and if the fide of one Square A,
meafure the fide of another B, the Jquare Aq ball likewife
meaſure the fquare Bq.
1. Hyp. For Aq: AB (a): AB: Bq, therefore ſeeing by
the hypothefis Aq meafures Bq, (b) it fhall meaſure alfo
AB. But Aq: AB:: A: B, (c) therefore alfo A meaſures
B. Which was to be demonftrated.
a 2.&11.8
b 7. 8.
c 20.def.7.
2. Hyp. A meaſures B; (c) therefore Aq fhall as well
meaſure AB, (c) as AB meafures Bq; (d) confequently Aq d 11. ax. 7.
meafures Bq. Which was to be demonftrated.
PROP. XV.
If a cube number
A, 2.
B, 6.
Ac meafures a cube
number Bc, then the
fide of the one (A) fhall meaſure the fide of the other (B.)
And if the fide A of one cube Ac measure the fide В of the
other Bc, also the cube Ac fhall meaſure the cube Bc.
Ac, 8. AqB, 24. ABq, 72. Bc,216.
1. Hyp For Ac, AqB, BqA, Bc (a) are; therefore
Ac, (6) meaſuring the extreme Bc, fhall alfo (c) meaſure the
fecond AqB. But Ac: AqB:: A: B, (d) therefore A
fhall alfo meaſure B. Which was to be dem.
2. Hyp. A meaſures B; (d) therefore Ac meaſures
AqB, which alfo meaſures BqA, and this Bc; (e) there-
fore Ac fhall meaſure Bc. Which was to be dem.
PROP. XVI.
If a ſquare number Aq do not mea-
fure a fquare number Bq, neither ſhall
the fide of the one A measure the fide
B, 9.
A, 4.
Aq, 16- Bq,81.
of the other B: And if A the fide of the one ſquare Aq do not
meaſure В the fide of the other Bq, neither shall the Square
Aq meaſure the fquare Bq.
1. Hyp. For if you affirm that A meaſures B, then
Aq alfo fhall meaſure Bq. Against the Hypothefis.
2. Hyp. If you maintain Aq to meaſure Bq; (a) then
likewife A fhall meaſure B. Contrary to the Hypothefis.
Hypothefis.
a 2.&12.8
b hyp.
c7.7.
d 20. def.7
e 11.ax. 7.
a 14. 8.
122
The eighth Book of
2 15. 8.
* 21. def. 7.
a 17. 7.
A, 2. B. 3.
Ac, 8. Bc, 27.
PROP. XVII.
If
a cube number Ac does not mea-
fure a cube number Bc, neither shall
the fide of one A meaſure the fide of
the other B: And if A the fide of one cube Ac does not mea-
fure В the fide of the other Вc, neither ſhall the cube Ac mea-
fure the cube Bc.
1. Hyp. Let A meaſure B; (a) then Ac all meaſure
Bc. Against the Hypothefis.
2. Hyp. Let Ac ineafure Bc; then A ſhall meaſure B ;
which is alſo against the Hypothefis.
C, 6. D, 2.
PRO P. XVIII.
CD, 12.
E,9. F, 3. DE, 18.
EF. 27.
*
Between two like plane num-
bers CD and EF there is one mean
proportional number DE: And
the plane C D is to the plane EF
in duplicate proportion of that
which the fide Chath to the homologous fide E.
For by the Hypothefis C: DE: F; therefore by
permutation, C: E::D: F. But C: E (a) :: CD : DE';
(a) and D: F:: DE: EF; (b) therefore CD: DE ::
c 10. def. 5. DE EF. (c) Wherefore the proportion of CD to EF is
duplicate to that of CD to DE,
tion of C to E, or D to F.
b 11.5.
ftrated.
Coroll.
that is, to the propor-
Which was to be demon-
Hence it is apparent, That between two like plane
numbers there falls one mean proportional in the pro-
portion of the homologous fides.
PROP. XIX.
CDE, 30. DEF, 60. FGE, 120. FGH, 240.
CD, 6. DF, 12. FG, 24.
C, 2. D, 3. E, 5. F, 4. G, 6. H, 10.
Betaveen two like folid numbers CDE, FGH, there are two
mean proportional numbers DFE, FGE. And the folid
CDE is to the folid FGH, in triplicats proportion of that
which the homologous fide C has to the homologous fide F.
Where-
EUCLIDE's Elements.
123
*
Whereas by the hyp. C:D :: F: G, and D:E::
G:H; therefore (a) by permutation, fhall CF::D:
G:: : C:
(a) E: H. But CD : DF (5) : : C : F, and DF
FG
(b): D:G; (c) wherefore CD: DF:: DF : FG::E:
H; (d) therefore CDE: DFE :: DFE : FGE::E: H
:: FGE: FGH. Therefore between CDE, FGH, fall
two mean proportionals DFE, FGE. (e) And ſo it is
plain that the proportion of CDE to FGH is triplicate to
that of CDE to DFE, or C to F. Which was to be dem.
Coroll.
Hereby it is manifeft, that between two like folid
numbers there fall two mean proportionals in the pro-
portion of the homologous fides.
PROP. XX.
If between two numbers A,
B, there falls one mean propor-
tional number C ; thofe num-
bers A, B, are like plane numbers.
A, 12. C, 18. B, 27.
D, 2. E, 3. F, 6. G, 9.
(a) Take D and E the leaft in the proportion of A to C,
or C to B, then D meaſures A equally as E does C; fup-
poſe by the fame number F; (b) alfo D equally meaſures
C, as E does B, fuppofe by the fame number G. (c) There-
fore DFA, and˜ÉG-B ; (d) and confequently A and B
are plane numbers. But becaufe EF (c)=C (c) = DG;
(e) therefore D:E: F: G, and alternately D:F::
E: G. (f) Therefore the plane numbers A and B are al-
fo like. Which was to be dem.
PROP.
If between two numbers
A, B, there fall two mean
proportional numbers C,D;
thofe numbers A, B, are
like folid numbers.
XXI.
A, 16, C, 24. D, 36. B,54.
E, 4. F, 6. G, 9.
H,z.P,2.M,4.K, 3.L,3.N,6,
* 21.def.7
a 13.7.
b 17. 7.
CII. 5.
d 17.7.
e 10.def. 5.
a 35.7.
b 21. 7.
€9.ax. 7.
d 16.def.7
e 19. 7.
f21.def.7.
a 2.8.
b 10. 8.
cz1.def.7.
d cor. 18.8.
(a) Take E, F, G, the leaſt in the proportion of A
to C, (b) then D and G are like plane numbers: let the
fides of this be H and P, and of that K and L; (c) there-
fore H:K :: P; L:: (d) E : F But E, F, G, (e) equally
meaſure A, C, D, fuppofe by the fame number M, and
likewiſe the ſaid numbers E, F, G, equally meaſure
the numbers C, D, B, fuppofe by the fame number, N.
()Therefore AEM=HPM, (ƒ) and B-GN-KLN; fg.ax. 7.
(g) and
e 21. 7.
124
The eighth Book of
g 17. def. 7, (g) and fo A and B are folid numbers. But becauſe C (f)
FN, therefore M: N ()
h 17.7.
k 7.5.
1 conft.
FM, and D (f)
): : FM :
FN (k): : C: D (/) : : E : F::H: K:
: E:F:: H: K : : P : L; (m)
therefore A and B are like folid numbers. Which was to
m 21. def. 7. be demonftrated.
a 19. 7.
bi.ax. 7.
€ 9. ax. 7.
>
AE, BF, CG, DH,
A, B, C, D,
E, F, G, H,
Lemma.
If proportional numbers A, B,
C,D, meaſure proportional num-
bers AE, BF, CG, DH, by the
numbers E, F, G, H, these num-
bers (E, F, G, H,) fhall be proportional.
For becauſe AEDH (a)
AEDH BFCG
(b) fhall
AD
BC
BFCG, (a) and AD=BC,
(c) that is, EH FG.
Which was to be demon.
(a) Therefore E:F::G: H.
Coroll.
Bq B B
X (e) For 1: B: B: Bq, (d) and
AqA A
d 15. def. 7.
Hence
B
B Bq
e lem. prec.
1 : A : : A : Aq, (d) therefore 1 :
A
A Aq,
(d) there-
Bq
B
B Bq
Bc..
X
fore
Aq
A A
Α' Α'
In like manner X
Ac Ac
Acc
3
and fo of the reſt.
PRO P. XXII.
Aq, B, C.
4, 8, 16,
If three numbers Aq, B, C, are continn-
ally proportional, and the first Aq a fquare,
the third C fhall alſo be a ſquare.
Bq
a 20. 7.
b
A q
7. ax. 7.
B
c cor. of the
lem prec.
d hyp. and
14. 8.
For becauſe AqC (a) =Bq, (b) thence is C =
B
But it is plain that is a number, (d) be-
A
(‹)=Q.= • But it is plain that
Bq
caufe
Aq,
A
or C is a number. Therefore if three, &c.
Ac, B, C, D,
8, 12, 18, 27.
PROP. XXIII.
J
If four numbers A, B, C, D, are con ·
tinually proportional, and the first of
them Ac a cube, the fourth aljo D,
all be a cube.
For
EUCLIDE'S Elements.
125
J
For becauſe Ac D (a) = BC, (4) therefore D =
B
Ac
BC
Ac
XC; that is, (becauſe A c C (d) Bq, and
(c) = (=C:
a 19.7.
b7. ax. 7.
ccor. of the
prec. lcm.
d 20. 7.
(c)=
Bq
(b) thence C=
Ac
!)D=
B
Bq
X
Ас Ac
Bc
Acc
B
Ac
B
Bc
But it is evident (e) that
is a number, becauſe
e 15.8.
Ac
Acc
or D is fuppofed a number.
&c.
Therefore if four numbers,
PROP. XXIV.
A, 16.
24. B, 36.
6. D, 9.
C, 4.
If tavo numbers A, B, are in the fame
proportion one to another, that a ſquare
number C is to a Square number D,
and the first A be a fquare number, the fecond alfo B ſhall be
a Square number.
Between C and D the fquare numbers, * and ſo be-
tween A and B having the fame proportion, (a) falls one
mean proportional. Therefore (b) fince A is a fquare
number, (c) B alſo ſhall be a ſquare number. Which was
to be demonftrated.
Coroll.
1. Hence, if there be two like numbers AB, CD, (A :
BCD) and the firft AB be a fquare, the fecond alfo
CD fhall be a ſquare.
* For AB: CD : : Aq : Cq.
2. From hence it appears, That the proportion of any
fquare number to any other not fquare, cannot poffibly
be refolved into two fquare numbers. Whence it cannot
be QQ:1 : 2, nor I Q: Q, &c.
::
nor 1: 5:
* 8.8.
a 11.8.
b byp.
C 22. 8.
11 and
18. 8.
PROP. XXV.
C, 64.
A, 8.
96. 144. D, 216.
12. 18. B, 27.
If two numbers A, B, are
in the fame proportion one to
another, that a cube number
C is to a cube number D, the first of them A being a cube
number; the ſecond В ſhall likerwiſe be a cube number.
(a) Between the cube numbers C and D, (5) and ſo be-
tween A and B having the fame proportion, fall two mean
proportionals; therefore (c) becauſe A is a cube, (d) fhall
B be a cube alfo. Which was to be demonſtrated.
Coroll.
2 12. 8.
b 8. 8.
c hyp.
d 23. 8.
126
The eighth Book of, &c.
* 12. and
19. 8.
a 18. 8.
b z. 8.
c cor. 2.8.
d 14. 7.
a 19. 8.
b 2.8.
€ 14. 7.
See Clavius.
Coroll.
1. Hence, If there be two numbers ABC, DEF, (A :
B:: D: E, and B: C:: E:F;) and the firft ABC be
a cube, the fecond DEF fhall be a cube alfo.
*For ABC : DFF:: Ac: Dc.
2. Hence it is evident, That the proportion of any
cube number to any other number not a cube cannot be
found in two cube numbers.
PRO P. XXVI.
A, 20. C, 30.
D, 4. E, 6.
B, 45.
F, 9.
ber is in to a fquare number.
Like plane numbers A, B,
are in the fame proportion one
to another, that a fquare num-
Between A and B (a) falls one mean proportional num-
ber C; (b) take three numbers D, E, F, the leaſt ÷ in
the proportion of A to C, the extremes D, F, (c) fhall
be fquare numbers. But by equality A: B(d): D:F;
therefore A: B::Q:Q. Which was to be demonftrated.
PROP. XXVII.
A, 16. C, 24.
D, 36. B, 54.
E, 8.
F, 12. G, 18. H, 27.
Like folid numbers
A, B, are in the
fame proportion one
to another, that a cube number is to a cube number.
(a) Between A and B fall two mean proportional num-
bers, namely, C and D (6) Take four numbers E, F, G,
H, the leaft in the proportion of A to C, (b) the
extremes E, H, are cube numbers. But A: B():
E:H:: C:C. Which was to be demonſtrated.
Schol.
1. From hence is inferred, that no numbers in pro-
portion fuperparticular, or fuperbipartitent, or double,
or any other manifold proportion not denominated from
a fquare number, are like plane numbers.
2. Likewiſe, that neither any two prime numbers,
nor any two numbers prime one to another, not being
fquares, can be like plane numbers.
The End of the eighth B o o K.
THE
[ 127 ]
The NINTH BOOK
O F
EUCLIDE's
ELEMENTS.
I
PROPOSITION I.
A, 6. B, 54.
Aq, 36. 108. AB, 324.
F two like plane numbers A, B, multiplying one another,
produce a number AB, the number produced AB ſhall
be a ſquare number.
For A: B(a):: Aq: AB; wherefore fince one mean pro-
portional (b) falls between A and B, (c) likewiſe one mean
proportional number ſhall fall between Aq and AB:
therefore fince the firft Aq is a fquare number, (d) the
third AB fhall be a fquare number alfo. Which was to
be demonftrated.
Or thus, Let ab, cd, be like plane numbers; name-
ly, a: b: c:d, (x) therefore adbe, and fo likewiſe
abcd or adbe, (y)= adad=Q: ad.
PROP. II.
If two numbers A, B, multiply-
ing one another, produce a fquare
number AB, thoſe numbers A, B,
are like plane numbers.
A, 6.
Aq, 36.
B, 54•
AB,324,
For AB (a): Aq: AB; wherefore fince between
Aq andAB, (b) there falls one mean proportional number,
(c) likewife one mean fhall fall between A and B, (d)
therefore A and B are like planes. Which was to be de-
monstrated.
a 17.7
b 18.8.
c 8.8.
d 22.8
x 19. 7.
y 1, ax. 7,
a 17.7.
br. 8.
c 8. 8.
d 2a. 8.
PROP.
128
The ninth Book of
2 15. def. 7.
b 17. 7.
c 8.8.
d 23. 8.
PROP. III.
A, z. Ac, 8. Acc, 64.
If a cube number Ac mul-
tiplying it felf produce a nuni-
ber Acc, the number produced Acc fhall be a cube number.
For, 1: A (a) :: A : Aq (6) : : Aq: Ac, therefore be-
tween 1 and Ac fall two mean proportionals. But 1: Ac
(a): : Ac: Acc; (c) therefore between Ac and Acc, fall
alfo two mean proportionals; and fo by confequence,
fince Ac is a cube (d) Acc fhall be a cube alfo.
Which was to be demomftrated.
Or thus; aaa (Ac) multiplied into it felf make aaaaaa
(Acc ;) this is a cube, whoſe fide is aa.
a 17.7.
b 12.8.
c 8.8.
dzz. 8.
Ac, 8. Bc, 27.
PRO P. IV.
Acc, 64. AcBc, 216.
If a cube number Ac mul-
tiplying a cube number Bc pro-
duce a number AcBc, the pro-
duced number AcBc fhall be a cube.
For Ac: Bc (a): : Acc: AcBc. But between Ac and
Bc (b) two mean proportional numbers fall; (c) therefore
there fall as many between Acc and AcBc. So that where-
as Acc is a cube number, (d) AcBc ſhall be ſuch alſo.
Which was to be demonftrated.
Or thus. AcBcaaa bbb (ababab)= C: ab.
a 17.7.
b 12.8.
c 8.8.
d 23.8.
Ac, 8.
Acc, 64.
B, 27.
PRO P. V.
AcB, 216.
If a cube number Ac mul-
tiplying a number В produce
a cube number AcB, the num-
ber multiplied В ſhall alſo be a cube.
For Acc: AcB (a): Ac: B. But between Acc and
ACB (5) fall two mean proportionals, (c) therefore alſo as
many ſhall fall between Ac and B, whence Ac being a cube
number, (d) B ſhall be a cube number alſo. Which
Which was
to be demonstrated.
PROP.
EUCLIDE's E'ements.
129
PROP. VI.
If a number A multiplying
A. 8. Aq, 64. Ac, 512.
it felf proluce Aq a cube, that
number A it felf is a cube.
For becaufe Aq (a) is a cube, and AqA (A c)(b) alſo a
cube; therefore () hdl A be a cube. Which was to
be demonftrated.
PRO P. VII.
If a compofed number A mul-
tiplying any number B, produce a
number AB, the number produced
AB fhall be a folid number.
A, 6. B, 11. AB,66.
D, 2.
E, 3.
Since A is a compofed number, (a) fome other number
▷ meaſures it, conceive by E, (b) therefore A = DE :
(c) whence DEB AB is a folid number, Which was to
be demonftrated.
PROP. VIII,
8
1. a, 3. a², 9. a³, z7. a¹, 81. a³, 242. a′, 729.
If from unity there are numbers continually proportional
how many foever (1, a, a², a³, aª, &c.) the third number from
unity a² is a fquare number; and ſo are all forward,
leaving one between (aª, a³, aº,&c.) But the fourth a³
is a cube number; and jo are all forward, leaving two be-
tween (a³, as, &c.) The feventh also a is both a cube
number and a fquare; and fo are all forward, leaving five
between (a¹², a's, &c.)
6
For 1. a²=Q:
:Q: a, and a4 = aaaa➡Q; aa, and as
aaaaaa Q: aaa, &c.
2. a3 aaa C. a, and a
@aaaaaaaa = C: aaa, &c.
_
aaaaaa C: aa,
: aa, and
a hyp.
big. def.
7.
€ 5.9.
a 13. def.7
bg. ax. 7s
C17. defi
b 20.7.
C 12. 8.
3. a6 —aaaaaa = C: aa = Q: aaa, therefore, &c.
Or according to Euclide; Becaufe 1:a (a): a: a", (b) a hyp.
fhall a'Q: a, therefore ſeeing a², a³, a*, are c) the
third a fhall be a fquare number; and fo likewiſe aº,
a³, &c. Allo becauſe ia (a): a: a³, therefore ſhall
a³, (b)—a² Xa = C: : a, (d) therefore the fourth from a³,
namely a", fhall be likewiſe a cube, &c. and confequent-
ly a6 is both a cube and a ſquare number, &c.
3
:
PROP.
d 23. 8.
}
130
The ninth Book of
a 22.8.
8.
b 23.
€ 20. 7.
d 3.9.
e 23.8.
PROP. IX.
1. a, 4. a², 16. a³, 64. aª, 256, &c.
1. a, 8. a², 64. a³, 512. a*, 4096.
If from unity
there are numbers
horv many foerer,
continually proportional (1. a², a³, &c.) and the number fol-
lowing unity, (a) be a ſquare; then all the reſt, aˆ, aˆ, a*,
&c. fhall be fquares also. But if the number next unity
(a) be a cube, then all the following numbers a², a³, a4, &c.
fhall be cube numbers.
1. Hyp. For a², a¹, e6, &c. are fquare numbers by
the preceding prop. alfo fince a is taken to be a fquare,
(a) therefore the third a³ fhall be a fquare, and likewife
a“, a¹, &c. and ſo all.
2
10
"
are
2. Hyp. a is put a cube, (b) therefore at, a", a
cubes; but by the prec. a³, a, a, &c. are cubes :
lafly, becauſe a: a: aa, (c) therefore ſhall a²-Q : a,
but a cube multiplied into it felf (d) produces a cube;
therefore a² is a cube, (e) and confequently the fourth
from it a; and in like manner a, a', &c. are cubes,
therefore all, &c. Which was to be demonftrated.
༨
More clearly thus. Let be the fide of the fquare
number a, and fo the feries a, a,² a³, a*, &c will
be otherwife expreffed, thus, bb, b7, b³, b³, &c. It
is evident that all theſe numbers are ſquares, and may
be thus expreffed, Q: b, Q: bb : Q: bbb, Q: bbbb, &c.
In like manner, if b be in the fide of the cube the
feries may be expreffed thus, b³, b6, b9, b¹², &c. or C :
b, C : b², C : b³, C : b+, &c.
PRO P. X.
1.95
1, a, n*, a³, at, a¹,
a
1, 2, 4, 8, 16, 32, 64,
a,
If from unity, there are
numbers how many foever con-
tinually proportional (1. a, a²,
a³, &c.) and the number next the unit (a) be not a square
number; then is none of the reft following a fquare num-
ber excepting a² the third from unity, and fo all forward,
leaving one between (a*, a6, a³, &c.) But if thai (a)
which is next after the unit, be not a cube number, neither is
other of the following numbers a cube, faving a³, the fourth
from the unit, and ſo all forward, leaving two between, a“,
a", a¹², &c.
any
2
1. Hyp.
EUCLIDE's Elements.
131
1. Hyp. For if it be poffible, let a be a fquare num-
ber; therefore becauſe a: a¹ (a): :a: a, and by inver-
fion, a³ : a+ :: a²;a; and alfo a' and a*(b) fquare
numbers, and the firft a² a fquare, (c) therefore a fhall
be likewife a fquare; contrary to the hyp.
2.
3
:a;
Hyp. If it may be, let at be a cube; fince (d) by
equality at: 26: :a: a³, and inverſely a6 : at::a
and alfo fince as and at are cubes, and the firſt a³ a
cube, (e) therefore a fhall be a cube alfo; against the
hypothefis.
PROP. XI.
If there are numbers how
many foever in continual
proportion from unity (8,a,
aˆ,
1. a, a²; a³, a²; a³, a²,
1, 3, 9, 27, 81,243,729.
a², a³, &c.) the lefs meafureth the greater by fome one of
them that are amongst the proportional numbers.
a::a: aa (a) therefore
aa
aaa
Becaufe a¦:a: aa
Alfo
aa
aaa
becauſe 1: aa(b): :a: aaa, therefore (a)
aa
a
at
a
aa
04
912 21
a
&c. Laftly becauſe 1: a² :: (b) a : 7, therefore
&C.
Coroll.
Hence, If a number that meaferes any one of pro
portional numbers, be not one of the faid numbers,
neither fhall the number by which it meaſures the faid
proportional numbers, be one of them.
1, a, a², a³, at
1,6, 36, 216, 1296.
B, 3.
PROP. XII.
If there are rumbers how many
foever in continual proportion from
unity (1,a,aª‚a', a+) whatsoever
prime number B, measure the laſt
at, the fame (B) fhall alfo measure the number (a) which
follow next after unity.
If you fay B does not meaſure a,(a)then 5 is prime to a ;
(b) and therefore B is prime to a²; (c) and fo confequently
to a*, which it is ſuppoſed to meafure. Which is abfurd.
I 2
Coroll.
a hyp.
bfuppof. &
8. 9.
C 24. 8.
d 14. 7.
e 25.8.
a 5. ax. 7:
&20.def.7:
b 14.7.
a 31. 7.
b 27.7.
c25.7%
132
The ninth Book of
3
a cor. 12. 9.
b 2. cor. 12.
9.
C 33.7.
e 3. cor. 12.
Coroll.
1. Therefore every prime number that meaſures the
laft, does alfo meaſure all thofe other numbers that pre-
cede the laft.
2. If any number not meaſuring that next to unity,
does yet meaſure the laſt, it is a compoſed number.
3.
If the number next to the unit be a prime, no
other prime number fhall meaſure the laft.
1, a, a², a³, at
PRO P. XIII.
1, 5, 25, 125, 625.
H--G--F- - E --
If from unity there are numbers
in continual proportion, how many
foever ( 1, a, a², a³, &c.) and that
after unity (a) a prime; then shall
no other meaſure the greatest uumber, but those which are
amongst the faid proportional numbers.
a,
there-
If it be poffible, let ſome other E meaſure a4, viz. by
F, () then F fhall be fome other different from a, a²
a³. But becauſe I meaſuring aª, does not meaſure a,
(b) therefore E fhall be a compoſed number, (c) therefore
fome prime number meaſures it, (d) which does confe-
quently meaſure at, (e) and fo is no other than
dii. ax. y. fore a meafures E. After the fame manner alfo may
F be fhewn to be a compofed number, meaſuring a*,
and to that a meaſures F. Therefore ſeeing EF (ƒ)=a*.
Confequently
axa³, (g) it will be a: E::F: a³,
whereas, a meaſures E, (b) likewife F fhall equally mea-
fure, viz. by the fame number G. (k) Nor ſhall G be
a, or a², therefore, as before, G is a compofed number,
and a meaſures it. Wherefore fince FG (ƒ)—a³—a² ×
(g) therefore a: F::G: a², and fo becauſe A meaſures
F, (b) G fhall equally meaſure a², viz. by the ſame num-
ber (k) which is not a. Therefore fince GH a²
و نوع
f
9. az. 7.
g 19. 7.
h 20. def. 7.
k cor. 11 9.
120.7.
a
2
ua() thence Hae: G, and becauſe a meaſures G
m 20. def. 7. (as before (H alfo fhall meaſure a, which is a prime
number. Which is imposible.
+
PROP,
EUCLIDE's Elements.
133
PROP. XIV.
If certain prime numbers B,
C, D, meaſure the leaft
number ▲, no other prime num-
ber E fhall meaſure the fame
befides thofe that meaſured it at first.
A
E
A, 30.
B, 2.
C, 3.
D, 5•
E--F--
If it is poffible, let be F; (a) then A= EF, (b)
therefore each of the prime numbers B, C, D, meaſures
one of thoſe E, F. Not E, which is taken to be a prime;
therefore F, which is less than A it felf; contrary to the
bypothefis.
PROP. XV.
A, 9. B, 12 C, 16.
D, 3. E, 4.
If three numbers "continually
proportional A, B, C, are the
leaft of all that have the fame
proportion with them; any two of them added together foall
be a prime to the third.
(a) Take D and E the leaft in proportion of A to B ; (b)
then A= Dq, and (b) C = Eq, (b) and B DE. But
becauſe D () is prime to E, (d) therefore fhall DE be
prime to both D and E,* therefore DX D--E (e)=Dq
+DE (A+B) (f) is prime to E, and fo to C or Eq.
Which was to be demonftrated.
(g) In like manner DE+ Eq (B+C) is prime to D,
and confequently to ADq. Which was to be demon.
ftrated.
Laftly, becauſe B (b) is prime to DE, it fhall
alfo be prime to the fquare of it (k) Dq + 2 DE -|- Eq
(A+ 2˜ˆB+C ;) (/) wherefore the faid B fhall be prime
to A+B+C, (4) and fo likewife to A+ C. Which
was to be demonftrated.
PROP. XVI.
If two numbers A, B, are
prime to one another, it ſhall not
A, 3, B, 5. C…….
be as the firft A, to the fecond B, fo is the fecond B to any
other C.
a 9. ax. 7.
b 32.7
a 35. 7.
b 2. 8.
c 24. 7.
d 30. 7.
* 26.7.
e 3. 2,
f as before
g 27.7.
h 26. 7•.
k 4.2.
1 30.7.
13
If
134
The ninth Book of
a 23. 7.
b 21.7.
c 6.ax. 7:
a 23. 7.
b 21. 7.
€ 20. def. 7.
d11. ax. 7.
29. ax. 7.
b 20.7.
£7.ax. 7.
If you affirm A: B:: B: C, then whereas A and B
(a) are the leaft in their proportion, A (b) fhall meaſure
B as many times as B does C; but A (c) meaſures it felf
alfo; therefore A and B are not prime to one another.
against the hypothefis.
PROP. XVII.
A, 8. B, 12. C, 18. D, 27. E---
If there are
numbers bow
many foever in continual proportion A, B, C, D, and the
extremes of them A, D, be prime one to another, the first A
fall not be to the fecond B, as the last D to any other E.
:
Suppofe A: B: D: E, then alternately A:D:
B: E, therefore feeing A and B are (a) the leaſt in their
proportion, A (5) ſhall meaſure B, (c) and B likewiſe C,
and C the following number D; () and fo A fhall mea-
fure the faid number D. Wherefore A and D are not
prime to one another; contrary to the hypothefis.
PROP. XVIII.
A, 4. B, 6. C, 9.
Bq, 36.
Two numbers being given A,
В, to confider if there may be a
third number C found proportional to them.
If A meaſures Bq by any number C, (a) then AC=Bq,
from whence (5) it is manifeft that A: B : : B : Č.
Which was to be done.
But if A does not meaſure Bq,
A, 6. B, 4. Bq, 16. there will not be any third pro-
portional. For fuppofe A: B
Bq
::BC; (a) then AC=Eq, (c) and confequently=C,
A
namely A meaſures Bq. Which is against the Hypothefis.
PROP. XIX.
A, 8. B, 12. C, 18. D, 27.
BC, 216.
Three numbers being
given A, B, C, to con-
fider if a fourth propor-
a 9. ax. 7:
tional to them D may be found.
If A meaſures BC by any number D, (a) then AD:
bax. 19. 7. EC; (5) therefore it appears that A: B
was required.
:C:D, which
EUCLIDE'S Elements.
135
But if A does not meaſure BC, then there can no fourth
proportional be found; which may be fhewn as in the
Îaſt prop.
PROP.
XX.
More prime numbers may
be gi-
A, 2.
B, 3. C, 5.
D, 30. G
-
ven than any multitude whatfo-
ever of prime numbers A, B, C,
propounded.
meaſure; If D
(a) Let D be the leaſt which A, B,
+ be a prime, the cafe is plain: if compofed, (5) then
fome prime number, fuppofe G, meaſures D+1, which
is none of the three A, B, C; For if it be, feeing it (c)
meaſures the whole D--1, (d) and the part taken away D,
(e) it ſhall alſo meaſure the remaining unit. Which is ab-
furd. Therefore the propounded number of prime num-
bers is increaſed by L1, or at leaſt by G.
PROP. XXI.
a 38. 7.
b
33.7.
c ſuppos.
d'conftr.
e 12.ax. 7.
5
5
3 3 2
2
A…………. E…………. B...F...C..G..D 20.
If even numbers, how many fever AB, BC, CD, are
added together, the whole AD jhall be even.
2
2
(a) Take EB = { AB, and FC= ½ BC, and GD=
CD; (5) it is plain that EB +FC+GD = { AD, (c)
therefore AD is an even number. Which was to be de-
monftrated.
a 6. dcf. 7.
b 12.7.
c 6 def. 7.
I
PRO P. XXII.
1
I
I
A……………….. F . B…………….. G. Ç..... H. D... L. E 22.
A......................... H.D... L.E
9
Zozu
5
3
If odd numbers, how many foever, AB, BC, CD, DE,
are added together, and the multitude of them be even, the
whole alfo AE fhall be even.
Unity being taken from each odd number, there will
(a) remain AF, BG, CH, DL, even numbers, (6) and
thence the number compounded of them will be even,
add to them the (c) even number made of the remaining
units, and the (d) whole AE will thereby be even.
Which was to be demonftrated.
a 7. def. 7.
b 21.9.
c hyp.
d 21. aa
I 4
PROP
.
136
The ninth Book of
a 22.9..
b 21.9.
€ 7. def. 7.
27. def. 7.
b hyp.
€ 21.9.
a 7. def. 7•'
b 24. 9.
7 def. 7.
PROP. XXIII.
7
5
I
A …………….. B ..……. C. . . E . D 15.
3
If odd numbers how
many foever, AB, BC,
CD, are added toge-
ther, and the multitude
of them be odd, the whole AD fhall be add.
For CD), one of the odd numbers. being taken away,
the aggregate of the others AC (a) is even. Whereto
add CD — 1, (b) the whole AE is alſo even; wherefore
the unit being reftored the whole AD (c) will be odd.
Which was to be demonftrated.
4
5
PROP. XXIV.
A....B.....D.C 10.
6
If an even number AB be
taken away from an even
number AC, that which re-
mains BC shall be even.
For if BD (BC-1) be odd, (a) BC (BD+1) will be
even. Which was to be demonftrated But if you fay BD
even, becauſe AC (b) is even, (c) thence AD will be
fo; (a) and confequently AC (AD+1) will be cdd, con-
trary the Hypothefis, therefore BC is even. Which was to
be demonftrated.
PROP. XXV.
6 I 3
A...... D. C... B 10.
7
If from an even number AB,
an odd number AC be taken
away, the remaining number
CB ſhall be odd
For AC—ı (AD) (a) is even (5) therefore DB is even ;
(c) and confequently CB (DB—1) is odd. Which was to
be demonftrated.
4
6
PROP. XXVI,
A..C... BII.
A.... Ç...... D. B 11.
If from an odd number AB
be taken away an odd number
CB, that which remaineth
AC stall be even.
Jor
EUCLIDE's Elements.
137
For AB 1 (AD) and CB-1 (CD) (a) are even;
a 7. def. 7.
(b) therefore AD-CD (AC) is even.
Which was to be
b 24. 9.
demonftrated.
PROP. XXVII.
I 4
6
If from an odd number
AB be taken away an even
number CB, the refidue AC
fhall be odd.
A.D....C……… . . . B 11.
5
...
For AB-1 (DB) (a) is even, and CB is fuppofed to be
even (b) therefore the refidue CD is even: (c) therefore
CD+1 (CA) is odd, Which was to be demonftrated.
PROP. XXVIII.
If an odd number A, multiplying an even
number В, produces a number AB, the number
produced AB fhall be even.
A, 3
B, 4
AB, 12.
For AB (a) is compounded of the odd number A taken
as many times as an unit is contained in B, an even
number. (b) Therefore AB is an even number. W.W.D.
Schol.
In like manner, if A be an even number, AB fhall be
an even number alfo.
PROP. XXIX.
If an odd number A multiplying an odd num-
ber B. produces a number AB, the number produ-
ced AB, fhall be odd.
A, 3-
B, 5.
AB, 15.
For AB (a) is compounded of the odd number B taken
as often as an unit is included in A, likewiſe an odd num-
ber. (b) Therefore AB is an odd number. Which was to
be demonftrated.
a 7. aef. 7.
b 24. 9.
€ 7. def. 7.
C
a hyp, and
15. def. 7-
b 21.9.
a15.def.7.
b 23.95
Schol.
1. An odd number A meaſuring an even
number B, meaſures the fame by an even
number C.
B, 12. (C, 4
A, 3.
For if C be affirmed to be odd, then becauſe (a) B≈
AC, (b) therefore B fhall be odd; against the hyp.
a.9 ax. 7.
b 29.9.
2. An
138
The ninth Book of
B, 15. (C, 5.
a 28.9.
a 28. 9.
A, 3.
2. An odd number A meaſuring an odd
number B, meaſures the fame by an
odd number C.
For if C be faid to be even, (a) then AC, or B will be
even, contrary to the Hypothefis.
B, 15. (C, 5.
A, 3.
For if either A or
3. Every number (A and C) that
meajures an odd number B, is it ſelf
an odd number.
C be affirmed to be even, B (a) fhall
be an even number, against the Hypothefis.
PROP. XXX.
B.24.
A, 3.
If
(C, 8.
D, 12.
A, 3-
(E, 4.
an odd number A meaſures an even number B, it ſhall
also measure the half of it D.
B
ȧ byp.
b. & Schol.
(a) Let
be=C, (b) then C is an even number.
A
29.9.
Therefore let E be C, then B (c) CA (d)=2 EA
€ 9. a. 7. (e) = 2 D, (f) therefore EA=D; (g) and confequently
dr. 2.
Y
Ꭰ
A
e hyp.
E. Which was to be demonftrated.
+ 60
£ 7. ax. F.
§ 7.ax. 7.
PROP. XXXI,
A, 5. B, 8. C, 16. D
a 3. Schol.
29. 9.
b. 3.0.9.
If an odd number A be
prime to any number B, it ſhall
allo be prime to the double thereof C.
If it be poffible, let ſome number D meaſure A and C,
(a) then D meaftring the odd number A fhall be odd it
felf, (b) and ſo ſhall meaſure B, the half of the even num-
ber C, therefore A and B are not prime one to another.
Which is against the Hypothefis.
Coroll.
It follows from hence, that an odd number which
is prime to any number of a double progreffion, is alfo
prime to all the numbers of that progreffion.
PROP.
EUCLIDE's Elements.
139
PROP. XXXII.
All numbers A, B, C, D, 1. A, 2. B, 4. C, 8. D,16.
c. in double progreffion
from trvo, are evenly even only.
It is evident that all thefe numbers A, B, C, D, (a) are
even, and (b), namely in a double proportion, (c) and
every leſs meaſures the greater by fome one of them.
(d) Wherefore all are evenly even.
But becauſe A is a
fo
prime number, (e) no number befides theſe ſhall meaſure
any of them. Therefore they are evenly even only.
Which was to be demonftrated.
PROP. XXXIII.
If of a number A, the half of B be odd,
the fame A is evenly odd only.
A, 30. B, 15.
D - E.
an
Since an odd number B (a) meaſures A by 2
even number, (b) therefore B is evenly odd. If you
affirm it to be evenly even, (c) then fome even number
D meaſures it by an even number E, whence 2 B (d) :
A (d) = DE; (e) therefore 2: E:: D: B, and there-
fore as 2 (f) meaſures the even number E, (g) ſo D an
even number meaſures B an odd. Which is impoffible.
PROP. XXXIV.
If
an even number A be neither doubled from
two, nor have it's half odd, it is both even-
ly even and evenly odd.
A, 24.
It is evident that A is evenly even, becauſe the half
of it is not odd. But becaufe, if A be divided into two
equal parts, and the half of it be again divided into two
equal parts, and if this be always done, we fhall at length
a 6. def. 7-
b2o.def. 7.
CII. 9.
d 8. def. 7.
e 13.9.
a hyp.
b 9. def. 7.
c 8. def. 7.
d 9.
9. ax. 7.
e 19. 7.
£ 6. def. 7.
g 20. def.
7.
fall upon fome (a) odd number, (for we cannot fall upon a 7. def. 7.
the number two, becauſe A is not ſuppoſed to be doubled
upward from two) which ſhall meaſure A by an even
number, for (b) otherwiſe A it ſelf fhould be odd; against b1.fch.29.
the Hyp.. Therefore A is evenly odd. Which was to be
demonstrated.
9.
1
PROP.
140
The ninth Book of
PRO P. XXXV.
A.....
.8.
4
8
B....F....
G 12.
C...
18.
6
8
9
4
N 27.
¿
a hyp.
B 17.5.
C 12. 5.
£ z. ax.1.
€ 18. 5.
D....
……. H…………… . L ....K….
If there are numbers in continual proportion how many
foever A, BG, C, DN, and the number FG, equal to the firft
A, be taken from the fecond, and KN, alfo equal to the first,
from the laft; it shall be as the excess of the fecond BF is to
the firft A, fo is the excess of the laft DK to all the numbers
that precede it, A, BG, C.
From DN take NL BG, and NH C. Becauſe
DN: C(HN) (a): : HN: BG(LN) (a): : LN (BG) : A
(KN;) (b) therefore by dividing every where, it will be
DH: HN:: HL: LN :: LK: KN; (c) wherefore DK:
C+BG+ALK (d) (BF): KN (A.) Which was
to be demonftrated.
Coroll.
Hence (e) by compounding, DN+BG+C: A- BG
+C: BG: A.
PROP. XXXVI.
1. A, 2. B, 4.
E, 31. G, 62. H, 124.
M, 31.
P.
C,8.
D, 16.
L, 248. F, 496.
N, 465.
Q---
a 14.7.
b 19.7.
€ byp
₫ 7. ax. 7.
e35.9.
£ 3. ax. 1.
14. 5.
If from unity be taken how many numbers foever 1, A, B,
C, D, in double proportion continually, until the whole added
together E be a prime number; and if this whole E multi-
ply'd into the laft D. produce a number F, that which is
produced F, Jhall be a perfect number.
Take as many numbers E, G, H, L, likewife in double
proportion continually; then (a) by equality A: D::
EL; (b) therefore AL DE (c) F, (d) whence L=
=F,
L, F, are in double pro-
F
2
Wherefore E, G, H,
portion. Let GE be
then M: E:: N:E+G+H+L. (f) But ME, (g)
M, and FEN; (e)
there-
EUCLIDE's Elements.
If I
h2.ax. 1.
k 7. ax.7.
111.ax. 7.
m11.9.
n 9. ax. 7.
0 19. 7.
P 13.9.
q 20.def.7.
r 31.7.
therefore NE+G+H+L; (b) therefore F1+ A
+B+C+D+E+G+HL=EN. More-
over becauſe D (k) meaſures DE (F)(1) therefore every one,
1, A, B, C, (m) meaſuring D, and (m) alfo E, G, H, L,
meaſures F. And further, no other number meaſures the
faid F. For if there does, let it be P, which let mea-
fures F by Q(n) therefore PQ=F=DE ; (e) therefore
E:Q:P: D, therefore feeing A a prime number, mea-
fures D, (p) and ſo no other P meaſures the fame, (q) con-
fequently E does not meaſure Q. Wherefore fince E is
fuppofed a prime number, (r) it ſhall be prime to Q (S)
wherefore E and Qare the leaſt in their proportion; (t)
and fo E meaſures P as many times as Q does D: (u) there-
fore Qis one of them A, B, C. Let it be B, feeing then
by equality BD:: E: H, (x) and fo BH-DE=F-
PQ, (x) and fo alfo Q: B::H: P, (y) therefore H=
P; therefore P is alſo one of them A, B, C, &c. Againſt
the Hypothefis. Therefore no other befide the forefaid
numbers meaſure F, and (2) confequently F is a perfect z 22. def 7
number, Which was to be demonſtrated.
{ 23. 7.
t 21. 7.
u 13.9.
x 19. 7.
y 14. 5.
The End of the ninth Book.
The
[ 142 ]
The TENTH BOOK
O F
EUCLIDE's
ELEMENTS.
Plate IV.
Fig. 10.
Fig. 11.
I
"M
Definitions.
Agnitudes are faid to be commenfurable
which one and the fame meaſure, meafures
them all.
B ;
The note of commenfurability is T, as A
that is, the line A of 8 foot is commenfurable to the
line B of 13 foot; becaufe D a line of one foot mea-
fures both A and B. Alſo √ 1
Alſo √ 1850, because
18 and √ 50. For √
9:
2 meaſures both ✔
으
​√
18:√
= 3 and √ 52 = √ 25 = 5, wherefore √ 18:
50: 3:5
II. Incommenfurable magnitudes are fuch, of which
no common meaſure can be found.
Incommenfurability is denoted by this mark; as ✔
6 √ 25 (5 ;) that is, √ 6 is incommenfurable to the
number 5, or to a magnitude defigned by that number
becauſe there is no common meaſure of them, as ſhall
appear hereafter.
10
III. Right-lines are commenfurable in power, when
the fame fpace meaſures their fquares.
The mark of this commenfurability is
; as AB
CD, i. e. the line AB of 6 foot is in power com-
menfurable to the line CD, which is expressed by V
20, becauſe E the space of one foot Square does as well
meaſure ABq (36) as the rectangle XY (20) to which the
Square of the line CD (20) is equal. The fame note
fometimes fignifies commenfurable in power only.
IV.
7
EUCLIDE's Elements.
143
IV. Lines incommenfurable in power are fuch,to whoſe
fquares no space can be found to be a common meaſure.
This incommenfurability is denoted thus; 50 √8. i. c.
5月
​the numbers or lines 5, and v 8 are incommensurable in
power, becauſe their ſquares 25 and 8 are incommen-
furable.
V. From which it is manifeft, that to any right-
line given, right-lines infinite in number are both
commenfurable and incommenfurable; fome in length
and power, others in power only. The right-line given
is called a Rational-line.
The note of which is §.
VI. And lines commenfurable to this line, whether
in length and power, or in power only, are alfo called
Rational
с
VII. But fuch as are incommenfurable to it, are called
Irrational,
And denoted thus p.
VIII. Alfo the fquare which is made of the faid given
right-line is called Rational, v.
C
IX. And likewife fuch figures as are commenfurable
to it, are Rational pa.
X. But fuch as are incommenfurable, Irrational
pa.
XI. And thoſe right-lines alfo, which contain them
in power, are Irrational p'.
Schol. Plate IV. Fig. 12.
That the last feven definitions may be rendered more clear by
an example, let there be a circle ADBP, whoſe femidiameter is
CB, infcribe therein the fides of the ordinate figures, as of a
Hexagon BP, if a triangle AP, of a ſquare BD, of a Penta-
gone FD. Therefore, if according to the 5. def the femidia-
meter CB be the Rational line given, exprejjed by the number
2, to which the other lines BP, AP, BD, FD, are to be com-
pared, then BP (a) =BC= 2, wherefore BP is BC,
according to the 6. def. Alfo AP (b)=y 12 (for ABq (16)
-BPq (4)=12) therefore AP is BC, likewife according
to the 6. def. and APq (12) is fœ by the o. def. Moreover
BD (b)=√/DCq+BCq=8; whence BD is BC; and
BDq pr. Lastly, FDq10-√20 (as hall appear by the
FDq—10—
praxis to be delivered at the 10. 13.) shall be a, according to
the 10. def. and confequently FD=√: 10~√20 is p, ac-
cording to the 11. def.
A Poftulate
a cor. 15. 4.
47. I.
b
144
The tenth Book of
a poſt. 10.
A Poftulate.
That any magnitude may be fo often multiplyed, till
it exceed any magnitude whatſoever of the fame kind.
Axioms.
1. A magnitude meaſuring how many magnitudes for
ever, does alſo meaſure that which is compofed of them.
2. A magnitude meafuring any magnitude whatſoever,
does likewife meaſure every magnitude which that mea-
fures.
3. A magnitude meaſuring a whole magnitude and a
part of it taken away, does alfo meaſure the refidue.
PROP. I. Plate IV. Fig. 13.
Two unequal magnitudes AB, C, being given, if from the
greater AB there be taken away more than half (AH) and
from the refidue (HB) be again taken more than half (HI)
and this be done continually, there fhall at length be left a
certain magnitude I B, less than the leſſer of the magnitudes
first given C.
(a) Take C fo often, till its multiple does fomewhat ex-
ceed AB, and let DF FGGEC. Take from AB
more than half AH, and from the remainder HB, more
than half, viz. HI, and fo continually, till the parts AH,
HI, IB, be equal in multitude to the parts DF, FG, GE.
Now it is plain, that FE which is not less than ½ DE, is
greater than H B, which is less than AB DE.
And in like manner GE, which is not leſs than FE, is
greater than IB HB, therefore C, or GE
Which was to be demonftrated.
IB.
The fame may alfo be demonftrated, if from AB the
half AH be taken away, and again from the refidue HB
the half HI, and fo forward.
PROP. II. Fig. 14-
Two unequal magnitudes being given (AB, CD) if the
lefs AB be continually taken from the greater CD, by an in-
terchangeable fubtraction, and the refidue do not meaſure the
magnitude going before, then are the magnitudes given in-
commenfurable.
If
EUCLIDE'S Elements.
145
If it be poffible, let fome magnitude E be the common
meaſure. Then becauſe AB taken from CD, as often
as it can be, leaves a magnitude FD lefs than it feff, and
FD taken from AB leaves GB, and ſo forward (a) there-
fore at length fome magnitude GB E fhall be left,
therefore E (6) meaſuring AB, (c) and fo CF, (b) and
the whole CD, (d) fhall alfo meaſure the refidue FD, (c)
confequently alfo AG; (d wherefore it fhall likewife
mcafure the remainder GB, less than it felf. Which is
abfurd.
1
PROP. III. Plate IV. Fig. 15.
Two commenfurable magnitudes being given AB, CD, to
find out their greatest common meaſure FB.
Take AB from CD, and the refidue ED from AB,
and FB from ED, till FB meaſure ED (which will come
to país at length, (a) becauſe by the Hyp. ABCD)
FB fhall be the magnitude required.
a i. io.
b bjp.
C 2. ax. 10.
d
d 3 ax.10:
á 2. 10.
is conft.
cz. ax. 10.
dı. ax. 10.
For FB (6) meafures ED, (c) and fo alfo AF; but
it meaſures it felf too, (d) therefore likewife AB, (c)
and confequently CE, (d) and fo the whole CD. Where-
fore FB is the common meaſure of AB, CD. If you
affirm Ġ to be a common meaſure greater than that,
then G meaſuring AB and CD, (e) meafures alfo CE
and (f) the remainder ED, (e) and fo AF; and (f) £3. ax. 10.
confequently the remainder FB, the greater the lefs.
Which is abfurd:
Coroll.
Hence, a magnitude that meaſures two magnitudes,
does alfo meaſure their greateſt common meafure.
PROP. IV. Fig. 16.
Three commenfurable magnitudes being given, A, B, C,
to find out their greatest common meaſure.
(a) Find out D the greatest common meafure of any two
A, B; (a) alfo E the greatest common meaſure of D and
C, therefore E is the magnitude fought.
(a) For it is clear, E meafuring D and C, (b) mea-
fures the three, A, B, C. Conceive another magnitude
F greater than that to meaſure them; (c) then F mea-
fures D, (c) and confequently E the greatest common mea-
fure of D, and C, the greater the lefs. Which is abfurd.
K
Corally
e 2.ax. 10.
a 3. io.
2 ax. 10.
b confir. &
c cor. 3.10.
146
The tenth Book of
Coroll.
Hence alſo it appears, that if a magnitude meaſures
three magnitudes, it fhall likewife meaſure their greateſt
common meaſure.
PROP. V.
+
'
A
C
B
D, 4.
F, I.
Commenfurable magni-
tudes A, B, have fuch pro-
E, 3.
portion one to another as
number hath to number.
a 3. 10.
b 20. def. 7.
€ 22.5.
a fch. 10. 6.
b conftr.
c byp.
d 22. 5.
c 5.ax. 7.
def. 7.
f 20.
g conft.
h 1. def. 10.
a 6. 10.
(a) C being found the greateſt common meaſure of A,
B; as often as C is contained in A and B, fo often is
I contained in the numbers I) and E; (b) therefore C : A
:: 1: D; wherefore inverſely A: C:: D: 1, (b) but
likewife C B : 1. E; (c) therefore by equality A: B
:: D:E:: N: N. Note, The letter N only fignifies
number in general, and refers not to any particular ſpace
or magnitude as the other letters do, and is to be
read, as A to B, fo D to E, and fo number to number.
Which was to be demonftrated.
E
HA
B
PRO P. VI.
F, 1.
C, 4.
D, 3•
If two magnitudes A,
B, have fuch proportion
one to another, as the num-
ber C hath to the number
D, thofe magnitudes A, B, fhall be commenfurable.
: 1: D.
What part is of the number C, (a) that let E be of
A. Therefore becauſe E: A (6) : :: C, and A: B (c)
::C: D, (d) therefore by equality fhall E : B :
Wherefore feeing 1 (e) meaſures the number D, (f) like-
wife Emeaſures B; but it (g) alfo meaſures A, (b) therefore
AB. Which was to be demonftrated.
PRO P. VII. Plate IV. Fig. 17,
Incommensurable magnitudes A, B, have not that proportion
one to another, which number hath to number.
If you affirm A:B:: N : N, (e) then À
the Hypothefis.
B, againſt
PROP.
EUCLIDE'S Elements:
147
PRO P. VIII. Plate IV. Fig. ij.
If two magnitudes A, B, have not that proportion one to
another, which number hath to number, thoſe magnitudes are
incommenfurable.
Conceive AB. (a) then A: B::N: N, contrary
to the Hypothefis.
PRO P. IX.
À
B
E, 4.
F, 3.
The fquares defcribed upon right
lines commenfurable in length, have that
proportion one to another, that a fquare
number bath to a fquare number. And
fquares, which have that proportion one to
another, that a fquare number hath to a ſquare number,
ſhall alſo have their fides commenfurable in length. But
fuch Squares as are made upon right- lines incommenfurable in
length, have not that proportion one to another, which a fquare
number hath to a Square number. And fquares which have
not fuch proportion one to another, as a quar: number hath
to a Iquare number, have not their fides commenfurable in
length.
1. Hyp. A B. I fay that Aq: Bq::Q:Q.
For (a) let A: B:: number E: number F; therefore
Aq A
Bq
(6) I
E
Eq
a 5. 10.
a 5. io.
b 20.6.
twice (c)= twice, (d)=Fq, (e) therefore Aq: cfcb.23.50
By: Eq: FqQ:Q. Which was to be demonftrated.
2. Hyp. Aq: Bq:: Eq: Fq: : Q: Q. Ifay AB. For
A
Aq
B
twice (ƒ) *(g) =
Bq
B: :
Eq
Fq
E
(3)= =twice, (i) therefore A :
E : F :: Ñ : N, (k) wherefore AB. Which was
to be demonftrated.
3. Hyp. A B
deny that Aq: Bq : : Q: Q. For
fuppofe Aq: Bq::Q: Q, then AB, as is fhewn be-
fore, against the Hypothefis.
4. Hyp. Not Aq: Bq: : Q: QI fay that AL B.
For conceive AB, then Aq: Bq :: Q: Q, as above,
against the Hypothefis.
Lines
lines
1.
Coroll.
are alſo, but not on the contrary. And
are not therefore, but Lines
K 2
are alfo
PROP.
di1.
d 11.8.
e 11.5.
f 20.6.
g hyp.
hii. 8.
ifch. 23.5.
É 6.10.
143
The tenth Book of
a 5. 10.
b 6. 10.
C 7. 10.
d 8. 10.
a ſch 10. 6.
b 3. 1.
a z. lem. 10.
10.
b 13. 6.
C 20. 6.
d conftr.
2.1. lem. 10.
10.
3.
b
10.
PROP. X. Plate IV. Fig. 18.
If four magnitudes are proportional (C:A::B:D)
and the first C be commenfurable to the fecond A, the
third B fhall be commenfurable to the fourth D. And if
the firft C be incommensurable to the fecond A, alfo the
third В fhall be incommenfurable to the fourth D.
If CA, (a) then Č: A:: N: Ñ, (b) : : B : D,
(b) therefore BD. But if CA, (c) then ſhall
not C: AN: N :: B; D, (e) wherefore B
Which was to be demonftrated.
Lemma 1.
D.
To `find out two plane numbers, not having the propor-
tion which a Square number hath to a Square.
Any two plane numbers not like, will fatisfy this
Lemma, as thoſe numbers which have fuper-particular,
fuperbipartient, or double proportion; or any two prime
numbers, See Schol. 27.8.
Lemma 2. Fig. 19.
To find out a line HR, to which a right-line given KM
bath the proportion of two numbers given B, C.
(a) Divide KM into as many equal parts as there are
units in the number B, and let as many of thefe, as there
are units in the number C, (b) make the right-line HR,
it is manifeft, that KM: HR: : B: C.
Lemma 3.
To find out a line D, to the fquare of which the fquare of
a right-line given KM hath the proportion of two numbers
given B, C.
Make BC (a): : KM: HR, and between KM and
HR, (b) find a mean proportional D. Therefore KMq:
Dq (e): KM: HR: B: C.
PRO P. XI. Fig. 20.
To find two right-lines incommenfurable to a right-line
given A, que D in length only, the other E in power alſo.
lem.. 10,
1. Take the numbers B, C, (a) fo that it be not B : C
:: Q: Q, (b) and let B: C:: Aq: Dq, (c) it is plain that
Q.,
D. But Aq (d) ´D Dq. Which was to be done.
∙C 9. 10.
d 6.10.
16.
2. (d)
EUCLIDE's Elements.
149
2. (d) Make A: EE: D. I fay Aq
Eq. For
d 13. 6.
A: Die): Aq: Eq, therefore fince AD, as before;
Eq. Which was to be done.
therefore Aq
PRO P. XII. Plate IV. Fig. 20.
Magnitudes (A,B) commenfurable to the fame magnitude C,
are alſo commenſurable one to the other,
Becauſe AC, and C B, (a) let A: C:: M:N
:: C: B:
D: E, and C : B :: M: N
D, 8. E, 8.
F, 2. G, 3.
F: G, (b) take three numbers H,
I, K, the leaft in the proportion
H, 5. I, 4. K, 6.
of D to E, and F to G. Now becaufe
A: C(): D: E(c) :: H: I, and C: B (c) :: F: G
:
::I: K, (d) therefore by equality, A: B::H: K::
M N, (e) therefore AB. Which was to be de-
monftrated.
e 20. 6.
f10. 10.
25.10.
b 4. 8.
© confir.
d zz. 5.
€ 6. xo.
Schol.
Hence, every right-line commenfurable to a rational
line is alfo it felf & rational. And all rational right-lines
are commenſurable to one another, at leaſt in power.
Alfo every ſpace commenfurable to a rational ſpace is
rational too: And all rational ſpaces are commenfurable
one to another. But magnitudes whereof one is rati-
onal, the other irrational, are incommenfurable amongst
themſelves.
PROP. XIII. Fig. 21.
If there are two magnitudes A, B, and one of them
A, commenfurable to a third C, but the other Bincom-
menfurable, thoſe magnitudos AB, are incommenfurable.
Conceive B A, then fince C (a) — A, (b) there-
fore CB, against the Hypothefis.
PRO P. XIV. Fig. 20.
If there are two magnitudes commenfurable A, B, and one
of them A incommenfurable to any other magnitude C, the
other alfo B fhall be incommenfurable to the fame C.
Imagine BC, then for that A (a)
fore AC, against the Hyp.
К
K 3.
12. 10. &
def. 6.
def. 9.
def. 7. and
10,
a hyp.
b 12. 10.
B, (b) there-
a hyp.
b12 10.
PROP.
150
The tenth Book of
a byp.
b 22. 6.
C 17. 5.
d 22. 6,
e cor. 4• 5:
£22. 5:
g 10. 10.
a 3. 10.
b 1.ax. 10.
c 1, def. 10.
d
3.ax. 10.
PRO P. XV. Plate IV. Fig. 22.
:
If four right-lines are proportional (A B C "D)
and the firft A be in prower more than the fecond В by the
fquare of a right-line commenfurable to it felf in length, then
alfo the third C, fhall be more in power than the fourth D
by the Square of a right-line commenfurable to it felf in
length. But if the first A be more in power than the
fecond B, by the fquare of a right-line incommenfurable to it
Self in length, then shall the third C be more in power than
the fourth D by the Square of a right-line incommenfurable
to it felf in length.
: C: D, (b) therefore Aq: Bq
:
For becauſe A ; B (a) :
:
:: Cq: Lq, (c) therefore by divifion Aq- Bq: Bq::
Cq-Dq: Dq; (d) wherefore √ Aq — Bq : B :
Bq : B : ✓
Cq- Dq: D, (c) and fo inverſely B: √ Aq~ Bq :
D:v (q_Dq, (f) therefore by equality A:
Aq—Bq::C:√ Cq-Dq, confequently if ATL, or
Aq-Bq, g) then likewife C, or
Cq-Dq. Which was to be demonftrated.
V
PROP. XVI. Fig. 23.
√
If two commenfurable magnitudes AB, BC, are compofed,
the whole magnitude AC fhall be commenfurable to each of the
parts AB, BC. And if the whole magnitude AC be com-
menſurable to either of the parts AB, or BC, theſe two mag-
nitudes given at first AB, EC, fhall be commenfurable.
1. Hyp. (a) Let D be the common meaſure of AF,
BC; (b) therefore D meaſures AC, and therefore AC IL
AB, and BC. Which was to be demonftrated.
2. Hyp. (a) Let D be the common meaſure of AC, AB,
(d) therefore D meaſures AC - AB (BC) and confe-
quently AB BC. Which was to be demonftrated.
Coroll.
Hence it follows, if a whole magnitude compofed
of two, be commenfurable to any one of them, the fame
fhall be commenfurable to the other alfo.
PROP,
EUCLIDE's Elements.
151
PRO P. XVII. Plate IV. Fig. 23.
If two incommenfurable magnitudes AB, BC, are compofed,
the whole magnitude allo AC fall be incommensurable to
either of the two parts AB, BC. And if the whole magnitude
AC be incommenfurable to one of them AB, the magnitudes
first given AB, BC, fhall be incommenfurable.
1. Hyp. If it can be, let D be the common meaſure
of AC, AB, (a) therefore
and therefore alfo AB
2. Hyp. Conceive AB
AB, against the hypothefis.
2 3. ax. 10.
D meaſures AC-AB (BC) (6) bi.def. 10.
BC, against the hypothefis.
BC, (c) therefore AC
L
c 16. 10.
Coroll.
Hence alfo, if one magnitude, compofed of two, be
incommenfurable to any one of them, the fame alſo ſhall
be incommenfurable to the other.
PROP. XVIII. Fig. 24.
If there are two unequal right-lines AB, GK, and upon
the greater AB a parallelogram ADB equal to the
fourth part of a square made of the less line GK, and
deficient in figure by a Square, be applied, and divides the
faid AB into parts commenfurable in length A D, DB; then
ball the greater line A B be more in pover than the lefs GK
by the fquare of a right-line FD commenfurable in length to
the greater. And if the greater AB be in power more than the
lejs GK, by the fquare of the right-line FD commenfurable
in length to it felf, and a parallelogram ADB equal to the
fourth part of the fquare made of the less line G K, and
deficient in figure by a Square, be applied to the greater AB,
then ſhall it divide the fame into parts A D, DB, com-
menfurable in length.
4
(a) Divide GK equally in H, and (b) make the rectan-
gle ADB GHq. Cut off AF-DB, then is ABq (c)=
ADB (d) (4 GHq or GKq) + FDq. Now in the first
place, if AD DB, then ſhall AB (e) BD (e) T
2 DB (ƒ) (AF+DB, or AB - FD) (g) therefore AB
FD. Which was to be dem. But fecondly, if ABFD,
(b) then ſhall AB AB FD ( 2 DB) (k) therefore
AB DB, (/) wherefore AD DB. Which was to
be demonftrated.
K 4
a 10. 1.
b 28.6.
c 8. 2.
d confir.&
4. 2.
e 16. 10.
f confir.
gcor.16.10.
hcor.16.10.
k 12. 10.
PROP. 116. 10.
152
The tenth Book of
217.10.
b 13. 10.
c cor. 17.
10.
d 13. 10.
€ 17. 10.
2
a 46. .
b 1.6.
c hyp.
d 10. 10.
e hyp. &
9. def. 10.
fiz. 10.
PRO P. XIX. Plate IV. Fig. 24,
If there are two right-lines unequal AB, GK, and to the
greater AB, a parallelogram ADB equal to the fourth part
of a fquare made upon the lefs GK, and deficient in figure
by a fquare be applied, and divides the faid AB, into parts
AD, DB, incommenfurable in length; the greater line AB-
Shall be in power more than the lefs GK by the fquare of the.
night-line FD incommenfurable to the greater in length, And
if the greater line AB be more in power than the lefs GK by
the fquare of a right-line FD incommenfurable to it in length,
and if alfo upon tho greater A B be applied a parallelogram
ADB equal to the fourth part of the fquare of the lefs
GK, and deficient in figure by a Square, then shall
it divide the faid greater line AB, into parts incommen-
furable in length AD, DB.
Suppoſe all the fame that was done and faid in the
prec. prop. Therefore firſt, If AD ■ D B, (a) then
fhall AB DB. (b) Wherefore AB 2 DB (AB--
FD) (c) therefore AB FD. Whith was to` be
monftrated.
de-
Secondly, If A BFD, then AB AB-FD
(2 DB) (d) wherefore AB DB, (e) and confequent-
ly AD DB. Which was to be demonftrated.
PROP. XX. Fig. 25:
A rectangle BD comprehended under right-lines BC, CD,
rational and commenfurable in length according to one of
the forefaid ways, is rational.
Let A be given p, and (4) the fquare BE defcribed up-
on BC. Becaufe DC: CE (BC)(b); ;BD: BE, and DC
(c) BC, (d) therefore fhall the rectangle B D be 11
fquare BE, wherefore feeing the fquare BE (e) Aq,
fhall alſo (ƒ) BD be Aq, and fo the rectangle pv BD .
Which was to be demonftrated.
Note, There are three kinds of rational-lines commen-
furable one to another. For either of two rational-lines
commenfurable in length one to the other, one is equal to the
rational-line propounded, or either of them is equal to it,
notwithstanding both of them are commenfurable to it in
length; or laftly both of them are commenfurable to the rati-
onal-line given only in power. And these are the ways which
the prefent Theorem Speaks of
+
In
EUCLIDE's Elements.
In numbers, let there be BC,
153
8 (2 √ 2) and CD
IZ.
18(3√ 2) then all the rectangle BD√ 144≈
PRO P. XXI. Plate IV. Fig. 26.
If
a rational rectangle D B be applied to a rational
Line DC, it makes the breadth thereof CB rational, and
commenfurable in length to that line DC whereto DB is
applied,
Let G be propounded f, and the íquare DA defcribed
on BC, becauſe BD: DA: : (a) BC : CA; and BD,
DA (6) are px (c) and fo (d) therefore BCTCA
but CD (CA) is p. (e) therefore BC is p. Which was to
be demonstrated.
In numbers, let there be the rectangle DB, 12, and
DC, 8. then ſhall CB, 18. but √/ 18=3√ 2. and
√8=2x+2.
Lemma.
To find out two rational right-
lines commenfurable only in
power.
Let A be proponnded §. (a)
A
B
C
21.6.
b byp.
cſch.12.10.
d 10. 10.
efch.12.10-
a
2 11. 13.
Take BA, (a) and CB, (b) it is clear that B bfch. 12.
and C are the lines required.
PRO P. XXII. Fig. 26.
A rectangle DB comprehended under the rational right-lines
DC, CB commenſurable in power only, is irrational: and
the right-line H, which containeth that rectangle in ponver
is irrational, and called a Medial-line.
Let G be the propounded f, and the fquare DA de-
fcribed on DC, and let Hq=DB. Becauſe AC: CB (a)
: : DA : DB, (b) and ACL CB, (c) fhall be DA
DB (Hq.) (d) but Gq DA; (e) therefore Hq Gq
(f) wherefore His p. Which was to be dem. and let it
be called a Medial-line, becauſe AC: H:: H: CB.
In numbers, let there be DC, 3. and CB, √ 6. then
fhall the rectangle be DB (Hq) 54. wherefore H is
✓ 54.
The note of a medial-line is μ, of a medial-rectangle
v, of both together μa.
Schol.
10.
a 1.
a 1. 6.
bhyp.
C 10. 10.
d hyp. and
9 def. 10.
e 13. 10.
f II. 10.
354
The tenth Book of
a ſch. 22.
10.
b 1. ax. I,
€ 14. 6.
d 22.6.
e hyp.
f fch. 12.
10.
g 10. 10.
h Sch. 12.
10.
k 1.6.
1 10. 10.
m fch. 12.
10.
n 13. 10.
0 1. 6.
p 10. 10.
a 11. 6.
b hyp.
€ 23. 10.
d 1.6.
e hyp.
10.
f 10.
g 12. and
13. 10.
h 22. 10.
Schol.
Every rectangle that can be contained under two ra-
tional right-lines commenfurable only in power, is me-
dial, although it be contained under two right-lines ir-
rational and every medial-rectangle may be contained
under two rational right-lines, commenfurable only in
power; as for example, the 24 is µ, becauſe it is con-
tained under 3, and 8, which are p,
✔ √
:
it may be contained under ✔ 6, and
for
although
96 irrationals,
24 576 = v √ 6 x ~ √ 96.
~
σχυν
PRO P. XXIII. Plate IV. Fig. 27.
If the rectangle BD made of a medial-line A, be applied
to a rational-line BC, it makes the breadth CD rational,
and incommenfurable in length to the line EC, whereunto the
rectangle BD is applied.
Becauſe A μ, (a) therefore fhall Ag be equal to fome
rectangle (EG) contained under EF and FG. (6)
therefore BD EG, (c) whence BC: EF :: FG: CD (d)
therefore BCq: EFq:: FGq: CDq. But ECq and LFq
(e) are pr. (f) and fo. (g) Therefore FGq CDq.
Wherefore fince FG is p, (b) therefore CI fhall be P.
Moreover, becauſe EF : FG (4) : : EFq: EG BD ;) for
fince EF FG, (e) fhall EFq be BD. But £Fq
(m) CDq. (n) therefore the rectangle BD a CDq.
Whence fince CDq : BD. (0) :: CD : BC. (p) ſhall CD
BC, therefore, &c.
be
PROP. XXIV. Fig. 28.
A right-line B commenfurable to a medial-line A, is alfo
a medial-line.
CD.
Upon CD p (a) make the rectangle CE=Aq; (a) and
the rectangle CF-Bq. Becauſe Aq (CE) is ur, (b) and
CD p, (c) therefore fhall the latitude DE be
But becauſe CE: CF(d): : ED: DF and CE (e)CF,
(f) therefore EDDF. (g) therefore DF is
whence the rectangle CF (Bq) is uv, and fo B is μ.
Which was to be demonftrated.
CD. (b)
Obf. that the note for the most part fignifies commen-
furable in pover only, as in this and the precedent demonftra-
tions, &c.
Coroll
EUCLIDE's Elements.
155
Coroll.
Hereby it is manifeft that a fpace commenfurable to a
medial-fpace, is alfo medial.
Lemma. Plate IV. Fig. 21.
To find out two medial right-lines A, B, commenfurable in
length and also two AC commenfurable only in power.
(a) Let A be any p, (b) take BA, and (c) C
A, (d) and 'tis evident the thing is is done.
PROP. XXV. Fig. 29.
•
A rectangle DR contained under DC, CB medial right-
lines commenfurable in length is medial.
Upon DC defcribe the fquare DA.
(DC) CB (a): : DA: DB, and DC
DA DB. (c) therefore DB is μr.
demonftrated.
Becauſe AC:
CB; (b) fhall
Which was to be
PRO P. XXVI. Fig. 30.
A rectangle AC comprehended under medial right-lines AB,
BC commenfurable only in power is either rational or medial.
Upon the lines AB, BC, (a) defcribe the fquares AD,
CE, and upon FG (b) make the rectangles FH =
AD, (b) and IKAC (b) and LM≈CE.
The fquares AD, CE, that is, the rectangles FH, LM,
(c) are μx and .. therefore GH, KM, having the fame
proportion (d) are ș, (e) and (f) therefore GHx KM
is pr. But becauſe AD, AC, CE, that is FH, IK, LM,
(g) are ; (b) and ſo GH, HK, KM alſo ÷; (k) thence
HKq GH × KM. (1) therefore HK is, or, or
没 ​IH (GF;) if, (m) then the rectangle IK or AC
is fv, but if(") then AC is uv. Which was to be dem.
Lemma. Fig. 31.
+
If A and E are only, Then first, fhall Aq, Eq, Aq
Eq, Aq-Eq (a). And fecondly Aq, Eq, Aq +
Eq, Aq-Eq AE and 2 AE. For A: E (6): Aq.
AE (6): AE: Eq. therefore feeing A (c) E, (d)
fhall Aq AE, (e) and 2 AE. Alfo Eq (d)
and 2 AE. wherefore becaufe Aq+Eq
and Aq-Eq
Aq and Eq; (f) therefore
a
a lem. 21.
10 and 13.
6.
b 2. lem.
10. IO.
c 3. lem 10.
10.
d conft. and
24. 10.
a 1. 5.
b 10. 10.
C 24. 10.
a 46. 1.
b cor.16.6.
c hyp, &
24. 10.
d 23. 10.
e 10. 10.
f 20. 10.
gfib.22.6.
h 1.6.
k 17. 6.
1 12, 10.
m 20. 10.
n 22. 10.
a hyp. and
16. 10.
b 1. 2.
AE, (e)
c byp.
Aq and Eq;
d 10. 10.
fhall Aq+
e 14. 10.
f 14. 10.
Eq, (f) and Aq-Eq beAE, and 2 AE, Hence
106
The tenth Book of
g 14. 10.
and 7. 10.
h cor. 7. 2.
a cor. 16.6.
b hyp.
€ 3. 10.
d
3. ax. 1.
e I. 10.
f 13. 10.
lem. 26.
10.
h ſch. 12.
10.
a byp.
b cor. 16.
10.
c ſch. 12. 10.
a fch. 12. 10.
b 16. 10.
cfch. 12. 10.
a lem. 21.10.
b 13.6.
€ 12. 6.
d 22. 10.
e conftr.
f 10. 10.
g 24. 10.
h 17.6.
h fch. 12.
10.
Hence alfo thirdly, Aq, Eq, Aq + Eq, Aq — Eq, 2
Aq + Eq + 2 AE; and Aq + Eq
AE (g)
AE. (g) and Aq+ Eq+ 2 AE
AE. (b) (Q. AE.)
2
Aq + Eq - z
PROP. XXVII. Plate IV. Fig. 32.
A medial rectangle AB exceedeth not a medial re&tangle
AC by a rational rectangle DB.
Upon EF ¿, (a) make EGAB, (a) and EH➡AC.
The rectangles AB, AC, i, e. EG, EH, (b) are μa; (c)
therefore FG and FH are EF. Whence, if KG,
(d) i. e. DB be pv, (e) then ſhall HG be HK; (f)
wherefore HG FH. (g) and confequently FGq
FHq. But FH is p. (b) therefore is FG p. but FG was
¿ before. Which is contradictory.
iş
Schol. Fig. 29, 33-
1. A rational rectangle A E exceeds a rational rectangle
AD by a rational rectangle CE.
For AE (a) AD, (b) therefore AE CE (‹)
wherefore CE is pv. Which was to be demonftrated.
2. A rational rectangle AD joined with a rational rect-
angle CF makes a rational rectangle AF.
*
For AD (a) CF, (b) wherefore AF AD and
CF; (c) and fo AF is pv. Which was to be demonftrated.
PROP. XXVIII. Fig. zz.
To find out medial-lines (C and D) which contain a
rational rectangle CD.
(a) Take A and B, (6) make A: C::C: B, (c)
and ABCD I fay the thing required is done.
For AB (Cq) (d) is pv, (d) whence Cis. But becauſe
ABCD (f) therefore CD; (g) and confe-
quently D is . Moreover by permutation A: C::
B: D). i. e. C: B:: B: D; (b) therefore Bq=CD.
But Bq is pv; (b) therefore CD is pv. Which was to be
done.
In numbers, let A be √ 2; and B √ 6. therefore C
is v 12. make √ 2: √6:: vy 12: D. or v4: ~ √
36::~√/12: D; then fhall D be v√ 108; but v√12X
~ √ 108 = √ √ 1296 =√ 36 =6. therefore CD is 6,
likewiſe C: D::13; wherefore C
D.
PROP
EUCLID E's Elements.
157
PROP. XXIX. Plate IV. Fig. 34-
To find out medial right-lines commenfurable in power only,
D and E, containing a medial rectangle DE.
=
:
μ.
(a) Take A, B, C, make A: D (6): D: B. (c)
and B : C : : D : E I lay the thing defired is performed.
For AB (d) Dq; and AB (e) is ur, therefore D is
μ; and B (f), (g) whence D E. (b) E is .
Moreover BC): D: E, and by permutation B :D
::: E, and by pe
GE. i. e. D: A: C: E; (2) therefore DEAC.
But AC (m) is ; therefore DE is μr.
Which was to
be done.
In numbers, let A be 20, and B,
I
200, and C, v 80.
Therefore D is + 80000; and E√ √✔ 12800._There-
alem21.10.
b 13. 6.
C 12. 6.
d 17. 6.
e 22. 10.
£ confir.
g 10. 10.
h 24. 10.
k conftr.
and cor 4-5•
fore DE=√14 1024000000
√32000. and D: E::
1 16. 6.
m 22. 6.
✔10: 2. wherefore D E.
Schol.
To find out two plane numbers, like
or unlike,
Take any four numbers pro-
portional A: B::C: Dit is ma-
nifeſt that AB and CD are like
plane numbers. And you may
find out as many unlike plane
numbers, as you pleafe, by the
help of Schol, 27.8.
A,
6.
B, 4.
AB, 24.
C 12
D, 8.
CD,
96.
C, 5.
D, 8.
CD, 40.
Lemma. Fig. 35-
A, 6.-
B, 4.
AB, 24.
AB, 24.
To find out two ſquare numbers (DEq and CDq) ſo that
the number compoſed of them (CEq) be fquare alfo.
Take AD, DB like plane numbers (of which let both
be even, or both odd) viz. AD, 24. and DB 6. The
total of theſe (AB) is 30; the difference (FD) 18. Half
of which (CD)is 9. (a) Now the like plane numbers AD, 218. 8.
DB, have one mean number proportional, namely DE;
therefore it is evident that every of thofe numbers CE,
CD, DE, are rational, and by confequence CEq (b)
(CDq+DEq) is the fquare number required.
Whereby it will be eafy to find out two fquare num-
bers, the excess of which is a fquare or not a fquare
number, namely by the fame conftruction (c) ſhall CÈq—
CDq be DEq.
But
b 47. 1.
C 3. ax. 1.
c3.
·158
The tenth Book of
a 24. 8,
b
cor. 24.8.
a 14. 5.
b 21. def. 7.
€ 26.8.
-
a 1. lem. 29.
10.
But if AD, DB be plane numbers unlike, the mean
proportional line (DE) fhall not be a rational number,
and fo neither fhall the excefs (DEq) of the fquare
numbers, CEq, CDq, be a fquare number.
Lemma 2.
2. To find out two fuch Square numbers B, C, as the
number compounded of them D is not fquare, Alſo to
divide a fquare number ▲ into two numbers B, C, not
Squares.
A, 3. B, 9. C, 36. D, 45.
1. Take any fquare number B, and let C bei B,
and DB + C. I fay the thing is done.
For B is Q, by the conftr. likewife becauſe B:C: : 1
:4::Q: Q: (a) therefore Calfo fhall be a fquare
number. But becaufe B+C(D): C:: 5:4: not
Q: Q. (b) therefore fhall not D be a fquare number.
Which was to be done.
A, 36. B, 24. C, 12. D, 3. E, 2. F, 1.
2. Let A be fome ſquare number.
plane numbers unlike, and let D be
DEA B. and D: F:: A: C.
required is done.
Take D, E, F,
EF. make
I fay the thing
=
For becauſe D: E+F ABC, and D E+
F, (a) therefore fhall A B+C. Now fuppofe B to
be fquare, (b) then A and B, (c) and confequently D and
E are like plane numbers. Which is contrary to the
Hyp.
The fame abfurdity will follow if C be fuppofed a
fquare number, Therefore, &c.
PRO P. XXX. Fig. 36.
To find out two rational right-lines AB, AF, commen-
Jurable only in power, fo that the greater AB shall be
in power,
more than the lefs AF, by the Square of a
right-line EF, commenfurable to it felf in length.
Let A B be p. (a) Take the fquare numbers CD,
CE, fo that CD' CE (ED) be not Q() and make
circle defcribed upon
the diameter AB (c) fit AF, and draw BF. Then I ſay
AB, AF, are the lines required.
b 3. lem. 10. CD : ED: : ABq: AFq. In a
io.
CI. 4.
}
For
EUCLIDE's Elements.
159
For ABq: AFq (d) : : CD : ED; (e) therefore ABq
AFq. But AB is p; (f) therefore AF is alfo p. But
becauſe CD is Q: and ED not Q: (g) therefore ſhall
AB be AF. Moreover by reaſon of the (b) right-
angle AFB, is ABq (k)= AFq+BP; therefore ſeeing
ABq: AFq:: CD: ED, by converfion of proportion
fhall ABq BFq:: CD: CE; : Q: Q. (1) Therefore
ABT BF. Which was to be done.
In numbers, let there be AB, 6; CD, 9 ; CE, 4 ; where-
fore ED, 5. Make 9: 5:36: (Q: 6.) AFq. then AFq
fhall be 20; and confequently AF 2c. Therefore BFq
36. — 20≈ 16. Wherefore BF is 4.
PROP. XXXI. Plate IV. Fig. 36.
To find out two rational lines AB, AF commenfurable
only in power, fo that the greater AB ſhall be in pozver more
than the lefs AF by the square of a right-line BF incom-
menfurable to it felf in length.
Let AB be p. (a) Take the fquare numbers CE, ED, fo
that CD = CE + ED be not Q, and in the rest follow
the conftruction of the preced. prop. I fay then the thing
required is done.
For, as above, AB, AF, are falfo ABq: BFq::
CD ED. therefore fince CD is not Q: AB, BF (¿) ſhall
be . Which was to be done.
In numbers, let there be AB, 5. CD, 45. CE≈ 36.
ED 9. Make 45 9:25 (ABq.):5 (AFq); therefore
AF=5. confequently BFq=452520. Wherefore
BF√20.
PROP. XXXII. Fig. 37.
To find out two medial-lines C, D, commenfurable only in
power, comprehending a rational rectangle CD, fo that the
greater C be more in power than the leffer D by the Square
of a right-line commenfurable in length to the greater.
(a) Take A and B ; fo as Aq - Bq T A (6)
and make A: C: : C: B. (c) and A: B:: C: D. I fay the
thing is done.
d conftr.
e 6. 10.
£ ſch. 12.
10.
g 9. 10.
h 31. 3.
k 47.
47. I.
19. 10.
a 2 lem. 29.
10.
b 9. 10.
a 30. 10.
(e) therefore fhall C.
b 13. 6.
C 12.6.
d conſtr.
For becauſe A and (d) B are
(f) (√ AB) be u. (g) and thence alfo CD () there-
fore D is likewife u. Furthermore, whereas A: B (d):
C: D; and by permutation A: C:: B: D: : C: B; and
Bq is pr; therefore fhall CD () (Bq) be pv. Laftly, be-
caufe Aq Bq (d) A, (1) fhall Cq- Dq be
/ T1 √ Ċq
C, therefore, &c. But if Aq - Bq Aq, then ſhall
√ Cq ~ Dq be C.
√
a
In
e 22. 10.
f 17.6.
g 10. 10.
h 24. 10.
k 17. 6.
1 15. 10.
160
The tenth Book of
a 30. 10.
b lem. 21.
10.
6.
·C13.
₫ 12.6.
e conftr.
f fch. 12. 10.
g 22. 10.
h. 10. 10.
£ 24. 10.
122. 10.
m 16. 6.
# 15.5.
a31 10.
b 10. 1.
€ 28. 6.
diz. 6.
e cor. 8.6.
£17.6.
£ 7. 5+
7.5¢
Fig. 10.
10. 10.
In numbers; let there be A 8, B√ 48 (√ : 64—16)
therefore C√ AB = √ √ 3072. and D
√3072. and D = v √ 1738,
wherefore CD √ 5308416√2304.
PROP. XXXIII. Plate IV. Fig. 38.
To find out two medial-lines D, E, commenfurable in
porver only, comprehending a medial rectangle DE, fo that
the greater D fhall be more in power than the lefs E, by
the fquare of a right-line commenfurable to the greater length.
(a) Take A and C, fo that √ Aq~Cq ri A, (b)
take alfo B A and C, and make A: D(c) :: D: B (d)
::C: E then D, and E are the lines fought for.
For becaufe A and C(e) are , (e) and BA and C,
(f) therefore ſhall B be, and D(AB) (g) fhall be u.
But becaufe A: D::C: E, therefore by permutation A:
C::DE. Wherefore feeing AC, therefore D fhall
be E; therefore E is μ Furthermore, (I) becauſe
D:B::C: E. and BC is µ ; alſo DE, equal to it, is īs
Laftly, becauſe A:C::DE(e) feeing √ Aq-CqZz
A; therefore ✓ Dq — Eqa D; therefore, &c. But if
Aq-Cq A. then / DqEq Eq.
In numbers, let there be A 8, С √ 48, B √ 28. Then
Dv ✔ 3072. and E v √ 588. wherefore D: E:: 2:
√ 3. And DE = √ 1344.
PRO P. XXXIV. Fig. 39.
To find ont two right-lines AF, BF, incommenfurable in
porver, whoſe ſquares added together make a rational figure,
and the rectangle contained under them medial.
(a) Let there be found AB, CD,; fo that ABG
-CDqAB; divide CD equally in G, (c) make the
rectangle AEB GCq. Upon AB, the diameter, draw
a femicircle AFB, erect the perpendicular EF, and draw
AF, BF. Theſe are the lines required.
a
For AE: BE (4) :: BA × AE : AB X BE. But BAX
AE (e)AFq; and ABX BE FBq. (f) Therefore AE:
EB: AFq: FBq; therefore fince AE (g) EB, (4)
AFq fhall be FBq. Moreover ABq (4) AFq + FBg
* 31. 3. (/) is p. Laftly EFq (AEB (4)CGq; (m) therefore
&
EF=CG, Therefore DE x AB — 2 EF X AB. But CD
x AB (7) is uv. (0) Therefore ABXED, EF, (p) or AF x
Which was to be demonſtrated.
FB is ..
k
47. I.
I conſtr.
m 1. ax. I.
1 22. 10.
24. 10.
pſch. 22.6.
.
The
!
EUCLIDE's Elements.
161
#
The Explanation of the fame by numbers.
4
3,
4
FB = √ 108,
Let AB be 6, CD 12; then CG=√¹²=√3. But
AE 36. and EB
6, whence AF
fhall be 18+✓ 216. and FB √ 18 ~ √ 216.
Alfo AFqFBq is 36, and AF x
But AE is found in this manner.
AF :: AF: AE. therefore 6 AE
(EFq) therefore 6 AE
then 18+ 6 e 9.
ce=6. wherefore e
√
Becauſe BA (6.):
A Fq AEq + 3
AEq 3. Put 3+e= AE.
6eee, that is, 9 -
6. and fo AE
ee
3+√√ 6.
PRO P. XXXV. Plate IV. Fig. 4C.
3. or
To find out two right-lines AE, EB, incommenfurable in
power, whofe fquares added together make a medial figure,
and the rectangle contained under them rational.
(a) Take AB and CF, fo that ABX CF be pv,
and ABq-CFq AB, and let the reft be done as
in the prec. prop. AE, EB are the lines required,
For, as it is fhewn there, AEqEBq. alfo ABq
(AEq-f· EBq) v. and laſtly AB x CF (b) is pv. (c) therefore
alfo ABX DE, that is, AE x EB, is py, therefore, &c.
PROP. XXXVI. Fig. 41.
To find out two right-lines BA, AC, incommenfurable in
power, whofe fquares added together make a medial figure,
and the rectangle alfo contained under them medial, and in-
commenfurable to the figure compofed of the fquares.
(a) Take BC and EF u, fo that BCX EF be fav.
and ✓ BCq-EFq BC, and fo forward, as in the prec.
BA, AC, fhall be the lines fought for
For (as above) BAq ACq, alfo BAq+ACq is ur.
end BAX AC is ur. Laftly, BC (6)
EF, and (c) fo
BC EG; likewife BC: EG (d): : BCq.: BCX EG
(BC X AD, or BAX AC) (e) therefore BCq (ABq +ACq)
BAX AC. therefore, &c.
Schol. Fig. 41.
To find out two medial-lines incommenfurable both in length
and porer.
a 32. 10.
b conft.
c fch. 12.
10.
dfch. 22.6.
a 33. 10.
b confr.
€ 13. 10.
dr.6.
€ 14.10.
1
L
(a) Take
S
162
The tenth Book of
a 36. 10.
b 13. 6.
€ 17.6.
d 14. 10.
a hyp.
b lem. 26.
10.
C 11. def. 10.
a hyp.
blem. 26.
10.
C II. def. 10.
a hyp.
b 21. 10.
a cor. 16.6.
b 47. 1. &
11.6.
c hyp.
d 16. 10.
e 24. 10.
(a) Take BC μ, and let BA x AC be
and let BA × AC be uv, and BCq
(BAq+ACq) (b) make BA : H::H: AC. then I ſay
BC and H are μ. For BC is u. (a) and BAX AC (c)
(Hq) is uv. wherefore H is alfo μ. (d) Likewife BA x AC
BCq; therefore Hq BUq. therefore, &c.
Here begins the fenaries of lines irrational by
compofition.
PROP. XXXVII. Plate IV. Fig. 42.
If two rational-lines AB, BC, commenfurable only in power,
are added together, the whole line AC is irrational, and is
called a binomial-line, or of two names.
For becaufe AB (a) BC, thence (b) fhall ACq be
ABq. But AB (a) is p. (c) therefore AC is p
Which was to be demonftrated.
PRO P. XXXVIII. Fig. 42.
If two medial-lines AB, BC, commenfurable in power
only, are compounded; and contain a rational rectangle, the
whole line AC is irrational and called a first bimedial-line.
For becauſe AB (a) BC, (b) fhall ACq be AB
× BC, pr. (c) therefore AC is . Which was to be demo.
—
Lemma. Fig. 43.
A rectangle AC, contained under a rational-line AB and
an irrational-line BC, is irrational.
For if the rectangle AC be affirmed pv, (a) then be-
cauſe AB is §, (b) the breadth BC ſhall be alſo §. againſt
the Hyp.
PRO P. XXXIX. Fig. 44.
If two medial-lines A B, BC, commenfurable only in
power, containing a medial rectangle, are compounded, the
whole line AC Jhall be irrational, and is falled a fecond
bimedial-line.
Upon the propounded line DE (a) make the rectangle
DF≈ACq; (b) and DGABq+BCq.
Becauſe ABq() BCq, (d) therefore ABq | BCq, i.e.
DG, ABq: but A3q(e) is us, (e) therefore DG is us.
But the rectangle ABC is taken ur, (c) and confequently
2 ABC
EUCLIDE's Elements.
163
p.
g 23. 10.
h lem. 26.
2 ABC (ƒ) (HF) is v. (g) therefore EG and GF are ș. f 4.2.
Alfo becauſe DG (½) HF; and DG: HF:: (k) EG :
GF; (1) therefore EGGF. (m) therefore the whole EF
is (2) wherefore the rectangle DF is pv. (o) therefore ✔
'DF, i. e. AC, is p, Which was to be demonftrated.
PROP. XL. Plate IV. Fig. 42.
If two right-lines AB, BC, commenfurable only in power,
are added together, making that which is composed of their
Squares rational, and the rectangle contained under them
medial, the whole right-line AC is irrational, and is called
a Major-line.
2 ABC
For whereas ABq+BCq (a) is pv, and (b)
(c) py; and fo ACq (d) (ABq + BCq+ 2 ABC) (e) —
ABq + BCq pv. (f) therefore ſhall AC be p. Which
was to be demonftrated.
PRO P. XLI. Fig. 45.
If two right-lines AC, CB, incommenfurable in power,
are added together, having that which is made of their fquares
added together medial, and the rectangle contained under them
rational, the whole right-line AB ſhall be irrational, and is
called a line containing in power a rational and a medial
rectangle.
For 2 rectangles ACB (a) fv, (b) — ACq +CBq (c)
pv. (d) therefore 2 ACB (d) ABq. wherefore (e) AB
is p. Which was to be demonftrated.
PRO P. XLII. Fig. 46.
If tavo right-lines GH, HK, incommenfurable in pover,
are added together, having both that which is compoſed of
their fquares medial, and the rectangle contained under them
medial, and incommenfurable to that which is compofod of their
Squares, the whole right-line GK is irrational, and is called
a line containing in power two medial figures.
Upon the propounded line FB make the rectangles
AFGKq, and CFGHq+HKq. Becauſe GHq-|-
HKq (CF) (a) is uv, the breadth CB (b) fhall be p. Alfo
becauſe 2 rectangles GHK (c) (AD) (a) is uv, therefore
AC (b) fhall be . Moreover becauſe the rectangle AD
(a). CF, (d) and AD: CF :: AC: CB, (e) thence
fhall AC be CB. (f) wherefore A is (g) p. therefore
the rectangle AF. i. e. GKq is ev; (b) and confequently
GK is p. Which was to be demonftrated.
L 2
PROP.
IO.
k 1. 6.
1 10. 10.
m 37. 10.
n lem. 38.
10.
0 11. def.
10.
a hyp.
b jch. 12.
10.
c hyp. and
24. 10.
d 4. 2.
e 17. 10.
f 11.def.10.
a hyp, and
ch. 12.10.
b fch. 12.
10.
c hyp.
d 17. 10.
e 11. def.
10.
a byp.
b 23. 10.
C 4. 2.
d 2. 6.
e 10. 10.
£ 37. 10.
g lem. 38.
10.
h11. def.
IO,
164
The tenth Book of
a 37. 10.
b ſch. 27.
10
c fch. 5. 2.
d fch. 12.
10.
€ 27. 10.
a 38. 10.
b sch. 27.
10.
c sch. 5. 2.
d 27. ic.
a 39.10.
b 16. and
24. 10.
C 23. 10.
d 24. 10.
€ 4. 2.
flem. 26. 10
g 1. 6.
h 10. 10.
k 37. 10.
PROP. XLIII. Fig. 47.
A line of tavo names, or binominal, AR, can at one
point only be divided into its names, AD, DB.
If it be poffible, let the binominal-line AB be divided
at the point E, into other names AE, EB. It is mani-
feft that the line AB is in both cafes divided unequally,
fince AD DB, and AE EB.
Becauſe the rectangles ADB, AEB (a) are pa; (a) and
each of ADq, DBq, AEq, EBq is p, (b) and fo ADq+
DBq (6) and AEq--EBq are alfo pa. (b) therefore ADG+
DBq : AEq +EBq (c) i. e. 2 AEB - 2 2 ADB is pv. (d)
therefore AEB ADB is pv. therefore exceeds
μν
fv. (e) Which is abfurd.
PROP. XLIV. Fig. 47.
Mv by
A first bimedial-line AB is in one point only D divide into
its names AD, DB.
Conceive AB to be divided into other names AE, EB,
whereupon every one ADq, DBq, EBq, will be (a) ua.
and the rectangles ADB, AEB, and the doubles of them,
pa; (b) therefore 2 AEB-2 ADB. (c) i. e. ADq -|- DBq
AEqEBq is pr, (d) Which is abfurd.
PRO P. XLV. Fig. 48.
A fecond bimedial-line AB, is divided into its names AC,
CB, only at one point C.
Suppoſe there were other names AD, DB. Upon the
propounded line EF make the rectangles EG ABq,
and EH ACq- CBq, as alfo EK ADg+ DBq.
Becauſe ACq, BCq (a) are μa ; (b) ACq + CBq
(EH) fhall be μv. (c) therefore the breadth FH is p. (a)
Moreover the rectangle ACB, (d) and fo 2 ACB (e) (IG)
is ur. (c) therefore HG is alfo p. And fince EH is (ƒ)
IG, (g) and EH: IG:: FH: HG. (b) therefore FH,
HG fhall be (k) therefore FG is a binomial, whofe
names are FH, HG. By the fame reafon FG is binomi-
al, and the names of it FK, KG: contrary to the 43 of this
Book.
PROP.
EUCLIDE's Elements.
165
PROP.
XLVI. Plate IV. Fig. 47.
A major-line AB is at one point only D divided into its.
names AD, DB.
Imagine other names AE, EB, whereupon the rectan-
gles ADB, AEB, (a) ua. (a) and as well ADq + DBq,
as AEqEBq are pr. (b) therefore ADqDBq — :
AEqEBq, (c) i. e. 2 AEB— 2 ADB is v. (d) Which
is imposible.
PROP. XLVII. Fig. 49.
A line AB containing in power a rational, and a medial-
figure is divided at one point only D into its names AD,
DB.
Conceive other names AE, EB, then both AEq-+
EBq, and ADq + DBq are ur. (a) and the rectangles
AEB, ADB are pr. (b) therefore 2 AEB 2 ADB, (c) i. e.
ADq + DBq, AEq + EBq is pv (d) Which is abfurd.
PROP. XLVIII. Fig, 48.
A line AB containing in power two medial-rectangles, is at
one point only C divided into its names AC, CB.
If you would have AB to be divided into other, names
AD, DB, draw upon the line compounded EF the
rectangles EG ABq, and EH = ACq+CBq, and
EH ADq+DBq. then becauſe ACq+CBq, name-
ly EH, (a) is µv, (b) the breadth FH fhall be . Alſo be-
cauſe 2 ACB, (c) that is, IG, is (a) pv, HG (6) ſhall be
likewife p. Therefore, whereas EH (a) IG, and EH
: IG (d): FH: HG, thence FH (e) fhall be HG,
(f) therefore FG is a binomial, and the names of it FH,
HG. In like manner FK, KG fhall be the names of it,
against the 43. of this Book.
A
Second Definitions.
Rational-line being propounded, and the binomial
divided into its names, the greater of whoſe names
is more in power than the lefs by the fquare of a right-
line commenfurable to the greater in length; then
L 3
I. If
a 40. 10.
b ſch. 27.
10.
csch.5.21.
d 17. 10.
a 41. 10.
b ſch. 27.
IO.
c ſch. 5.2.
d 27. 10.
a 42. 10.
b 23. 10.
C 4. 2.
di. 6.
e 10 10.
£37. 10.
166
The tenth Book of
I. If the greater name be commenfurable in length to
the rational-line propounded, the whole line is called a
binomial-line.
II. But if the leffer name be commenfurable in length
to the rational-line propounded, the whole line is called a
fecond binomial.
III. If neither of the names be commenfurable in
length to the rational-line propounded, it is called a
third binomial.
Furthermore, if the greater name be more in power
than the lefs, by the fquare of a right-line in-
commenfurable to the greater in length, then
IV. If the greater name be commenfurable to the
propounded rational-line in length, it is called a fourth
binomial.
V. If the leffer name be fo, a fifth.
VI. If neither, a fixth.
PROP. XLIX.
A....4 C..... 5 B
D-
a fch. 29. 10.
E-
•F
G
b 2 lem. 10.
H-
10.
c 3 lem. 10.
IO.
d confir.
e 6. def. 10.
£ 6. 10.
gfch. 12.
10.
h 9. 10.
kg, 10.
I
1 1 def.48.10.
To find out a first binomial-
line, ÉG.
(a) Take A B, AC, fquare
numbers, whofe excefs CB is
not Q. let 1) be propounded f.
EFq :
(6) Take EFD, and (c) make A B CB
FGq. then EG fhall be (a) firft binomial.
For EF (d) D. (e) therefore EF is p. (f) alfo EFq
FGq. (g) therefore FG is alſo f. likewife (d) becauſe
EFq: FGq::AB: CB:: Q: not Q(6) therefore EF
FG. Laftly, becauſe by converfion of proportion, EFq:
EFqFGq;:AB: AC::Q:Q thence EF (k) fhall
be✓ EFq FGq. ( therefore EG is a firft bino-
mial. Which was to be done.
In nnmbers thus ; let there be D 8. EF 6. AB 9. CB
5. wherefore becauſe 9:5: : 36: 20. therefore F G is
√20.
20. and confequently EG is≈ 6+ √ 20.
A……….4C…………. 5.B
D
E
F
G
H-
Prove it as
the prec.
PROP. L.
To find out a fecond binomial-
line, EG.
Take A B and A C ſquare
numbers, the excess of which is
CB not Q. Let the line D be
D, and make CB AB::
propounded f. take FG
FGq: EFq, then EG will be the line deſired.
:
For
EUCLIDE's Elements.
167
:
For FGD. wherefore FG is p. Alfo EFqFGq
therefore EF is p. Likewife becaufe FGq: EFq: CB:
AB: not QQ, thence FG is EF. Laftly, ſeeing
CB: AB:: FGq: EFq, and inverſely AB: CB:
FGq. therefore as in the foregoing Prop. EF
FGq. (a) whereby EG is a fecond binomial.
was to be done.
EFq :
/ EFq
Which a 2 dof.48.
In numbers; let there be D 8, FG 10, AB 9, CB 5;
then EF is 180, wherefore EG is 10+√ 180.
IO.
To find out a third binomial
line, DF.
PROP. LI.
A.... 4 C………..5 B
L.
6.
G
afch. 19.
E-F 10.
D
H.
(a) Take AB, AC, fquare
numbers, the excefs of which
CB is not Q, and let L be
a number not Qnext greater
than CB, viz. by a unit or two.
pounded § (6) Make L:AB:: Gq: DEq, (b) and AB:
CB:: DEq: EFq, then DF ſhall be a third binomial.
Let G be the line pro-
For becauſe DEq (c) □ Gq, (d) DE is p. alfo Gq:
DEq:: L:AB:: not Q: Q. (e) therefore G DE.
Likewife fince DEq (e) EFq, (d) alfo EF is p. More-
over becauſe DEq: EFq:: AB: CB :: Q: not Q, (f)
is DE EF. and fince by conftr. and equality Gq: EFq
::L: CB:: not Q: Q. (for (g) L and CB are not like
it plane numbers) (b) therefore fhall G be alfo EF.
Laftly, as in the prec. prop. ✔ DEq-EFq
therefore DF is a third binomial. Which was to be
done.
DE. (4)
In numbers; let there be AB, 9. CB, 5. L, 6. G, 8.
then ſhall be DE √ 96, and EF√ 4°. wherefore DF
= √96 + √ +0.
480
9
b 3 lem. 10
ΙΟ
cconftr. 6.
10.
d fch. 12.
10.
e 6. 10.
f 9. 10.
g ſch.27.8.
h 9. 10.
k 3def. 48.
10.
To find out a fourth binomial-line
DF.
PRO P. LII.
A... 3 C………….. 6 B
G
afch. 29.
D
E-F
10.
H-
(a) Take any ſquare numberAB,
and divide it into AC, CB not
fquares. Let G be the line pro-
pounded §. (b) take DE G, (c) and make AB: CB b 2. lem. 10
:: DEq: EFq, then DF fhall be a fourth binomial.
10.
L 4
C
c 3 lem. 10.
For
10.
168
The tenth Book of
d g. 10.
e 4. def.48.
10.
a 9. 10.
b
19.
5. def. 48.
{
For, as in the 49 of this Book, DF may be fhewn to
be a binomial, and alſo becauſe by conſtr. and converſion
of proportion DEq: DEq- EFq: AB: AC: :Q;
not Q. (d) fhall DE be✓ DEq. -EFq (e) therefore
DF is a fourth binomial.
In numbers, let G be 8. DE, 6. then EF fhall be
24. therefore DF is 6+ √ 24.
A... 3 C......6 F.
G
D
H
E- F
ed p take EF
PROP. LIII.
DF
To find out a fifth binomial-line,
Take any fquare number AB
whofe fegments AC, CB are not
Q. Let G be the line propound-
G. and make CB: AB::EFq: DEq.
then ſhall DF be a fifth binomial.
For DF ſhall be a binomial as in the 50. of this Book,
and becauſe by conſtruction, and inverfion, DEq : EFq
:: AB; CB, and fo by converfion of proportion, DEq:
DEq EFq:: AB : AC::Q: not Q(a) therefore
fhall DE be✓ DEq— EFq. (b) therefore DF is a
fifth binomial. Which was to be done.
PROP. LIV.
A……….. 5 C…………. . . 7 B
G
Da
H
L.
.9
E—— F
To find out a fixth bino-
mial-line.
Take A C, CB, prime
numbers, fo that A C+CB
(AB) be not Q. take alfo any
number ſquare L. Let G be
the line propounded f. (a) and make L:AB:: Gq:
DEq, and AB: CB:: DEq: EFq. then DF fhall be a
a 3. lem. 10. the line propounded
[O.
bfch. 27. 8.
cg. 10.
d6 def. 48.
10.
fixth binomial.
For DF may be demonftrated binomial as in the 51.
of this Book, and alſo by reaſon that DE and EFG.
Laftly likewiſe becauſe by conſtr. and converſion of pro-
portion DEq: DEq EFq: AB AC :: not Q: Q
(For AB is prime to AC, (6) and fo unlike to it) () there-
fore DE DEqEFq, (d) therefore DF is a fixth
binomial Which was required.
be
In numbers, let there be G 6. DE ✔ 48. then EF ſhall
28. wherefore DF is 48+ 28.
Lemma.
EUCLIDE's Elements.
169
Lemma. Plate V. Fig. 50.
Let AD be a rectangle, and the fide thereof AC divided
unequally in E; alfo let the lefer portion EC be equal-
divided in F. Upon the line AE (a) make the rectangle
AGE EFq, and from the points G, E, F, (b) draw GH,
É,
EI, FK, parallel to A B. (c) Let the fquare LM be made
equal to the rectangle AH, and upon OMP produced the
fquare MNGI, and let the right-lines LOS, LOT,
NRS, NPT, be produced.
I fay 1. MS, MT, are rectangles. For by reafon of
the right-angles of the fquares OMQ, RMP, (a) fhall
QMR be a right-line. (b) Therefore RMO, QMP, are
right-angles, wherefore the parallelograms MS, MT, are
rectangles.
2. Hence it is plain that LS (c)
ly that LN is a ſquare.
a 28.6.
b31, 1.
C 14. 2.
afch, 15.1
b 13. 1.
LT, and confequent c 2. 4. 1.
For
3. The rectangles SM, MT, EK, FD, are equal.
becauſe the rectangle AGE (d) EFq. (e) thence fhall
AG: EF:: EF: GE, (f) and fo AH: EK: : EK : GI,
that is by conftr. LM: EK: : EK: MN. (g) but LM :
SM:: SM: MN; therefore EK (4) ≈ SM (4)
=SM (4) FD
(1):
(/)= MT.
= AD.
=
4. Hence LN (m):
5. Becaufe EC is equally divided in F, (n) it is plain that
EF, FC, EC are 1.
√
6. If AE EC, and AE AEqECq, (0)
then fhall AG, GE, AE, be . alfo becauſe AG: GE
:: AHGI, (p) therefore fhall AH, GI, ¿. e. LM, MN,
be . Likewiſe
7. OM MP. For by the Hyp. AEEC (9)
therefore EC GE (9) wherefore EF GE. but EF
:GE:: EK: GI (r) therefore EKGI that is, SM
MN. But SM: MN::OM: MP. (r) therefore OM
MP.
8. If AE be fuppofed
d hyp.
e 17. 6.
f 1. 6.
gfch.22.6.
h9.5.
k 36. 1
143. 1.
m 2 ax. I.
n 16.
16. 10.
o 18. and
16. IC.
p 10. 10,
q 14. 10.
10. 10.
✓ AEq-ECq, it is appa- f19. and
whence (/) LM MN.
rent that AG, GE, AE, are
for AG: GE : : AH : GI :: LM : MN.
Thefe being well confidered, we fall easily dispatch the fix
following Propofitions.
17. 10.
PROP.
170
The tenth Book of
PROP. LV. Plate IV. Fig. 50.
If a ſpace AD be contained under a rational-line AB, and
a firft binomial-line AC (AE -- EC) the right-line OP
which containeth that ſpace in power is irrational, and called
a binomial-line.
All that being fuppofed which is defcribed and de-
monftrated in the foregoing Lemma, it is manifeft that
the right-line OP containeth in power the ſpace AD. (a)
Likewife AG, GE, AE, are L. therefore feeing AE,
(b) is AB, (c) fhall alfo AG and GE be AB.
(d) Therefore the rectangles AH, GI, that is, the fquares
LM, MN, are pz. therefore OM, MP are p¸ (e), (ƒ)
and confeqently OP is a binomial. Which was to be
e lem. 54.10. demonftrated.
a byy, and
lem. 54.10.
b byp.
c fch. 12. 10.
d 20. 10.
37.10.
"
a hyp. and
lem. 54. 10.
b hyp.
cfch. 12.
In numbers, let there be AB5. AC4 -- √ 12. wherc-
fore the rectangle AD = 20 + √ 300 to the fquare
LN. Therefore OP is 15+5. namely a fixth
binomial.
PRO P. LVI. Fig. 50.
If a ſpace AD be comprehended under a rational line AB,
and a fecond binomial AC (AE+EC) the right-line OP,
which containeth that Space AD in power, is irrational, and
called a first medial-line.
P
The forefaid Lemma of the 54 of this Book being
again fuppofed, then fhall OP be = √ AD. (a) alfo
AE, AG, GE, are . therefore fince AE (b) is
AB, likewiſe AG, GE (c) ſhall be § AB, therefore
the rectangles AH, GI, i. e. OMq, MPq. (d) are pa, (e)
10. Moreover OM MP. Laftly, EFL EC, and EC
(ƒ) AB. (Wherefore EK, i, e. SM, or OMP, is
e lem. 54.10.
pv, (b) Confequently OP is a firft bimedial. Which was
₤ hyp. 12.
d 22. 10.
70.
g 20. 10.
h 38. 10.
to be dem.
In numbers, let there be AB 5, and AC √ 48: +6.
then the rectangle AD: 1200+ 30 OPq. there-
fore OP is v√675 to√ 75, viz. a firſt bimedjal.
PROP. LVII. Fig. 50.
If a fpace AD be contained under a rational-line AB, and a
third binomial-line AC (AE+EC) the right-line OP which
containeth in pover the ſpace AD, is irrational, and called a
Second bimedial-line.
As
EUCLID E's Elements.
171
As above, OPqAD. Alfo the rectangles AH, GI,
that is, OMq, MPq area (a) Likewiſe EK, or OMP
is r. (5) Therefore OP is a fecond bimedial.
600
In numbers, let there be AB 5. AC √ 32 + √ 24.
wherefore AD is √ 800 + √ OPq. and fo OP is
~ √ 450 +~√ 50, that is, a fecond bimedial.
PROP. LVIII. Plate IV. Fig. 50.
If a ſpace AD be comprehended under a rational-line AB,
d a fourth binomial AC (AE+EC) the right-line OP
containing the Space AD in power, is that irrational-line
which is called a Major-line.
For again, OMq (a) MPq; and the rectangle AI,
i. e. OMq+ MPq (b) is pr. (c) alfo EK or OMP is uv.
(d) therefore OP (√ AD) is a Major line. Which was
to be demonftrated.
In numbers, let there be AB 5. and AC 4+ √ 8.
then the rectangle AD is 20 + 200. wherefore OP
is √:20 + √ 200.
PRO P. LIX.
If a space AD be contained under a rational-line AB, and
a fifth binomial AC, the right-line OP which containeth the
Space AD in power, is that irrational-line, which is a line
containing a rational and a medial rectangle in power.
Again OMP MPq. and the rectangle AI or OMq
+MPq is μv. (a) Likewiſe the rectangle EK or OMP is
v. (b) therefore OP (√ AD) contains in power & and
Which was to be demonftrated.
In numbers, let there be AB 5, and AC 2 -
the rectangle AD = 10 +√200 = OPq.
OP is √ 10 + √ 200.
PROP. LX.
μ.
8; then
Wherefore
If a space AD be contained under a rational-line AB and a
fixth binomial AC (AE +EC) the line OP containing the
Space AD in power is irrational, which containeth in power
two medial-rectangles.
As often before, OMq MPq and OMq+ MPq
is ur. and alfo the rectangle (EK) OMP is ur. (a) there-
fore OP√ AD contains in power 2 μa.
Which was
to be demonftrated.
In numbers, let there be AB‍5, AC √
therefore the rectangle AD or OPq is √ 300
and fo OP is√ √ 300 + √ 200,
12 + √ 8,
+ √ 2000
Lemma
a hyp, and
22. 10.
b 39. 10.
a lem. 54.
IO.
b hyp, and
20. IO.
c hyp, and
22. 10.
d 40. 10.
a as in the
prec.
b 41. 10.
a 42. 10.
172
The tenth Book of
a 4. 2. and
3. ax. 1.
b 7.2.
€ 1.6.
d 16. 10.
Lemma. Plate IV. Fig. 51.
Let a right-line AB be unequally divided in C, and let
AC be the greater fegment, and upon fome line DE apply
the rectangles DF = ABq, and DHACq, and IK ==
CBq, and let LG, be divided equally in M, and alſo MN
drawn parallel to GF.
I fay, 1. The rectangle ACB is
For 2 ACB LF.
LN or MF. (a)
2. DLLG. for DK (ACq+CBq) (b) ► LF (2
ACB therefore fince DK, LF are of equal altitude, (c)
DL fhall be LG.
CB, (d) then fhall the rectangle DK be
3. If AC
ACq and CBq.
4. Also DL
e lem, 26.
jo.
i. e. DK
f 10. 10.
g 1.6.'
h 17.6.
k hyp.
1 10. 10.
m 18. 10.
n 19. 10.
a hyp.
blem. 60.
10.
therefore DL
LG. For ACq+CBq (e) 2 ACB,
LF, but DK: LF (e):
LG.
DL: LG, (ƒ)
:
5. Moreover DL √ DLq - LGq. For ACq: ACB
(g): ACB CBq. that is DH: LN: IK, ( LN:
wherefore DL: LM : : LM : IL; (b) therefore DI × IL
LMq. therefore feeing ACq (4) CBq, that is, DHT
IK, and (/) fo DI IL, (m) ſhall DL be
LGq. Whith was to be demonftrated.
DLq -
6. But if ACq be put CBq, (n) then ſhall DL be
√ DLq-LGq.
This Lemma is preparatory to the fix following Propa-
fitions.
PROP. LXI.
The Square of a binomial-line (AC+CB) applied unto
a rational-line DE, makes the latitude DG a firft binomial-
line.
Thoſe things being fuppofed, which are defcribed
and demonftrated in the preceding Lemma, becaufe AC,
CB (a) are p, (b) the rectangle DK fhall be a
c fch. 12. 10. ACq; (c) and to DK is ; (d) therefore DL DE ș:
But the rectangle ACB, and fo 2 ACB (LF) (e) is pay.
(ƒ) therefore the latitude LG is DE, (g) therefore
alfo DL LG alfo DL / DLqLGq. From
whence (k) it follows that DG is a firſt binomial.
Which was to be demonstrated.
d 21. 10.
e 22. and
24. 10.
£ 23. 10.
g 13, 10.
h lem. 60.
10.
k 1 def. 48.
.
C
10.
PROP.
EUCLIDE's Elements.
173
PRO P. LXII.
The fquare of a first bimedial-line (AC+CB) being ap-
plied to a rational-line DE, makes the latitude DG a fecond
binomial-line.
The aforefaid Lemma being again fuppofed; The rec-
tangle DK ACq; (a) therefore DK is v; (b) there-
fore the latitude DL is DE. But becauſe the rec-
tangle ACB, and fo LF (2 ACB) (c) is ¿v, (d, fhall
“
a 24. 10.
b 23. 10.
; (f)
c hyp, and
Sch. 22. 10.
d 21. 10.
LG be DE; (e) therefore DL, LG are
&
alfo DL DLq LGq; (g) from whence it is
clear that DG is a fecond binomial. Which was to be
demonftrated,
PROP. LXIII.
The fquare of a fecond bimedial-line (AC+CB) applied
to a rational-line DE makes the breadth LG a third bino-
mial-line.
As in the prec. DL is DE. Furthermore becauſe
the rectangle ACB, and fo LF (2 ACB) (a) is μv; (b)
therefore ſhall LG be DE. (c) Moreover DL
LG. and alfo DL√ DLq —LGq; (d) therefore DG
is third binomial, Which was to be demonftrated.
a
م
PRO P. LXIV.
The fquare of a Major-line (AC+CB) applied to a ra-
tional-line DE, makes the breadth DG a fourth binomial-
line.
Again ACq+CBq. i. e. DK (a) is pv, (b) therefore
DL is DE, alfo ACB, and fo LF (2 ACB) (c) is
μv; (d) therefore LG is DE, (e) and confequently
DLLG. Laftly, becauſe AC BC, (f) fhall DL
beDLqLGq, (g) whence DG is a fourth bino-
mial. Which was to be demonftrated.
PROP. LXV.
The fquare of a line containing in power a rational and a
medial rectangle (AC+ CB) applied to a rational-line DE
makes the latitude DG a fifth binomial.
Again, DK is v; (a) therefore DL is ' DE; alſo
LF is pv; (b) therefore LG is DE; (c) therefore
DL LG; (d) likewife DL √ DLq. LGq; (e)
and fo by confequence DG is a fifth binomial. Which
was to be demonſtrated.
e 13. 10.
f lem. 60.
10.
gz.def.48.
10.
a byp. and
24. 10.
b 23.10.
clem. 60.
10.
d3. def. 48.
10.
a hyp, and
ſch. 12. 10.
b 21. 10.
c hyp, and
24. 10.
d
23. 10.
e 13. 10.
flem. 60.
10.
84. def.48.
10.
a 23. 10.
b 21. 10.
C13.10.
d lem. 60.
10.
PROP.
e5. def.48.
10.
174
The tenth Book of
a hyp.
b 14. 10,
CI. 6.
d 10. 10.
e lem. 60.
10.
£ 6. def. 48.
10.
PROP. LXVI.
The Square of a line containing in power trvo mediul rec-
tangles (ABCB) applied to a rational line DE, makes
the latitude DG a fixth binomial-line.
с
As before, DL and LG are DE. But becauſe
ACq+CBq (DK) (a)' ACB, (b) and fo DK LF
(2 ACB) and alſo DK : LF (c) : : DL : LG. (d) therefore
fhall DL be LG. (e) Laftiy DL✓ DLqLGq;
(f) by which it appears that DG is a fixth binomial.
Lemma.
A
C
B
A
B
C
D—
-F
E
D-
-EF
a 19.5.
b 1.6.
c before ·
d 10. 10.
e zz. 6.
£ 10. 10.
g 10. 10.
Let AB, DE be . and make AB : DE::AC: DF.
I fay 1. ACDF, as appears by 10. 10. alfo CB u
FE; (a) becauſe AB: DE: : CB : FE.
2. AC : CB::DF:FE. For AC: DF: :AB: DE
::
:: CB FE; therefore, by permutation, AC: CB::
DF: FE.
3. The Rectangle ACS DFE. For ACq: ACB (6)
:: AC:CB(c): : DF:EF: :DFq: DFE, wherefore by
permutation ACq: DFq: : ACB: DFE, therefore fince
ACq DFq, (d) fhall ACB be DFE.
4. ACq + CBq DFqFEq, For becaufe ACq
: CBq. (e): DFq: FEq; therefore by compounding
ACq-f-CBq : CBq:: DFq+FEq: FEq, therefore fince
CBq FEq, (ƒ) ſhall alfo ACq+CBq be DFq,
+FEq.
5. Hence, If AC or CB, (g) then likewiſe
fhall DE be or EF.
PROP. LXVII.
a lem. 66.
10.
b hyp.
clem. 66.
10. and
fch. 12. 10.
d 15. 10,
A line DE, commenfu-
A-
D-
rable in length to a binomial
line (AC+CB) is it felf a
binomial-line, and of the fame order.
CB
F
E
Make AB DE:: AC: DF; (a) then are AC, DF
(a) and CB, FE; whence fince AC and CB, (6)
are, thence, DF, FE; therefore DF is a
binomial. But becauſe AC: CB (a) :: DE: FE. If
ACIL OF √ ACq — BCq, (4) then in like manner
DF
EUCLIDE's Elements.
175
T
orp.
or a
or propound-
or
p.
e 12. 10.
and 14.10.
f by def.48
BF or √ DFq— FEq; alfo if AC
DF
propounded, (e) then fhall DF be
ed. But if CB or p. likewife FE
If both AC, CB,, (f) then alfo both DF, FÉ,
(g) That is, whatſoever binomial AB is, DE ſhall
be of the fame order. Which was to be demonftrated.
PROP. LXVIII.
A line DE commenfurable in length to a bimedial-line (AC
+CB) is alſo a bimedial-line, and of the fame order.
Make AB: DE:: AC: DF; (b) therefore AC
DF; and CBEF; therefore ſeeing AC and CB, c) are
μ, (d) alfo DF and FE fhall be ; and becaufe AC (e) I
CB, (e) therefore FD FE. (f) therefore DE is 2.
Wherefore if the rectangle ACB is p ; becauſe DFE
(b) ACB, (g) likewiſe DFE is py; and if that be µv,
(b) this fhall be μy too. (k) That is, whether AB be 1
bimed, or 2 bimed, DF fhall be of the fame order. Which
was to be dem.
PROP. LXIX.
A
A line DE commenfura-
ble to a Major-line (AC †- D
CB) is it felf a Major-line.
:
:
C
B
F-
-E
10.
g 14. 10.
a 12.6.
b Lem. 66.
10.
c byp.
d 24. 10.
e 10 10.
f 38. 10.
gſch. 12.
IO.
h 24. 10.
k 38. or 39
10.
b lem. 66.
10.
Make AB DE AC: DF. Becaufe AC (a) a hyp.
CB, (b) thence DF FE; alfo ACq+ CBq (a) is pr;
and fo becauſe DFq — FEq (6) □ ACq + CBq ; (c)
alfo DFqFEq is py; laftly, the rectangle ACB (a) is
v; (d) therefore the rectangle DFE is g (becauſe
DFE is (6) A C B) (e) wherefore DE is a Major-
line. Which was to be demonftrated.
PROP. LXX.
A line DE commenfuarble to a line containing in power a
rational and a medial-rectangle (AC+ CB) is a line contain-
ing in power a rational and a medial-rectangle.
Again make AB: DE: AC: DF. Becauſe AC (a)
CB, (b) alfo DF FE; likewife becauſe ACq ť
CBq (a) is µv (c) therefore DFqFEq fhall be μ;
laftly becauſe the rectangle ACB (a) is pv, (d) alfo DFE
is pv. Therefore DE contains in power fy and Which
was to be dem.
fev.
cfcb.12.10.
24. 10.
e 40. 10.
a hyp.
blem. 66:
10.
C 24. 10.
d ſch.12.10
PROP.
A
176
The tenth Book of
{
PROP. LXXI.
L
A
Ꭰ
C.
F
B
E
a hyp.
b lem. 66.
10.
C 24. 10.
d 24. 10.
e 14. 10.
f 42. 10.
b 2. ax. 1.
C 21. 10.
❤
A line DE commenfurable to
a line containing two medial-rec-
tangles in power (AC+CB) is
alfo a line containing in power two medial-rectangles.
Divide DE, as in the prec. Becaufe ACq (a) CBC,
(b) thence fhall DFq be FFq; alfo becauſe ACq+
CBq (a) is pv, (c) fhall DFq - FEq be alfo vy. And in
like manner becaufe ACB (2) is v, (d) alfo DFE is u.
Laftly, becauſe ACq+CBq ACB, (e) fhall DFq -†-
FEq be DFE. (f) From whence it follows that DE
contains in power 2 μz.
Which was to be demonftrated.
PROP. LXXII. Plate V. Fig. 1.
If a rational-rectangle A and a medial B, are compounded,
four irrational-lines will be made; either a binomial, or a
first bimebial or a Major, or a line containing in power a
rational and a medial-rectangle.
Namely, If Hq≈ A+B, then H ſhall be one of the
four lines which the Theorem mentions. For upon
a cor. 166. CD the propounded f, (a) make the rectangle CEA,
and FIB (b) and fo CI=Hq. Whereas then is A p,
likewife, CE is pv. (c) therefore the latitude CF is
CD, and becauſe B is uv, alfo FI fhall be pr; (d) there-
fore FK is CD, (e) therefore CF, FK are .
and fo the whole CK (f) is binom. wherefore is AB,
i.´e. CE — FI, (g) then CFFK. therefore if CF
V CFq-FKq; (b) likewiſe CK fhall be a 1 bin. and
confequently HCI (k) is a bin. If CF be fuppoſed
d 23. 10.
€ 33. 10.
f 37. 10.
1.6.
g 1.
ĥ 1. def. 48.
10.
k 55. 10.
14. def. 48.
10.
m 58. 10.
n 2.def. 48.
10..
o 56. 10.
P 5. def. 48.
· 10.
919. 10.
√ CFq- FKq, (4) then ſhall CK be a 4 bin. where-
fore H (CI) (m) is a major-line. But if AB, (g)
then fhall CF be FK. confequently if FK / FKq
-CFq, (2) then fhall CK be a 2. bin. () wherefore H
is a firft 2. Laftly if FKFKq-CFq, (p) then
CK fhall be a fifth binom. (g) whence H fhall contain in
power pv and v. Which was to be demonßrated.
ov
PRO P. LXXIII. Fig. 1.
If two medial rectangles, A, B, incommenfurable to one
another are compofed, two remaining irrational-lines are
made, either a fecond bimedial, or a line containing in porwer
two medial rectangles.
1
As
Plate IV. Facing Pag.176.
Fig. 1.
H
D
NE
KT Fig. 2.
Fig.3.
HF QG
F
G M
I
K
Fig. 4. Fig 5.
D
F
L
MNGVI
D
D
GH
N
R
P
I
PN
A
K B
EZ B R
I
S
B C
F DEA F
A
P
Fig. 6. A
Fig/7.
E
Fig. 10.A
B Fig. B
Xx
Fig 8.
R
Fig.
I
A
136
Leld
20
H
20
N
Y
BD
E
B
M
D
F
Fig. 13
B
B
AF
D
Fig. 15.
B
B
A Fig. 10.
Fig. 18.
Fig.17.
PUKE
Big
Fig. 19
C
K+
+M
H
R
FFig 12.
F
T B.
B
-D-
F
A
-F
TGT
C
-E-
-F-
ABD
E
P
Fig 14.
Fig. 26.
A
Fig.23.
C
B
ED F
CH
A
Br
A
Fig 20
ABC
D
AC D ACE
A Fig. 21.
Fig.27.
+
B
Fig. 24.
A Fig. 22.
D
A
B
Fig.26.
FC D
AVA
B
F D
E
G
DH
Fig. 29.
E
H K
FILN
D
B
31
C
E C DA C BGH
Fig. 37.A
E
B C
A
A
Fig.28.
Fig. 30
B
E
AYM
B
E
A
G HKM
B
CE
F
F
D
E
Fig 35.
Fig./36.
F
KD Fig. 32
F D
E
Fig 34.
A
Fig. 33.
D
F C
B
A
B
F
Fig. 38
D
H
A
B
Fig.37.
A DBCE
AH
BH
AARCH
Ar
E..... D
F
E
D
Fig. 41.
Br
CH
DH
Fig.42
AF B
G
B
A
D
-B
H
-C
Fig. 39.E B
Fig.45.C
Fig. 40.
D
E
B B
D
D CEH
F
B
Fig. 44.
A Fig 430
Fig. 46.
E
F
Ꮐ
H
K
Fig. 47.
A
C
B
AC
FEDB
A
Fig. 49.
B
B
A
E
PCB
F
E D
R
N
D
G
LI
A
TC
LM
EF
P
M
Fig. 51.
F
F
KH
B
HI K D L
A E
H K N
Fia. 48.
Fig. 50.
*
الله
EUCLIDE's Elements.
177
As H containing in power A+B is one of the faid
irrational lines. For upon CD propounded make the
rectangle CE A, and FiB whenice Hq = CI.
Therefore becauſe CE and FI (a) area, (b) the latitudes
CF, FK, fhall be CD. Alfo becaufe CE (a) I
FI, and CE: FI (c) :: CF : FK, (d) therefore CF
FK, (e) therefore CK is a bin. namely, if CF TL √
CFq-FKq, whence H✓ CI (f) fhall be 2 μ. But if
CF ✓ CFq - FKq, (g) then CK fhall be a 6 binom.
(4) and confequently H contains in power 2
was to be demonfirated.
a. Which
a hyp
b 23. 10.
c 1. 6.
C
d 10. 10.
e 3. def. 48.
JO
f 57. 10.
g6 def. 48.
10.
Here begins the Senaires of lines irrational by h 60. 10.
Subtraction.
PROP. LXXÍV.
If from a rational-line DF a D-
rational-line DE, commenfurable
EF
in power only to the whole DF, be taken away, the refidue
EF is irrational, and is called an Apotome or refidual-line,
For EFq (a) DEq; (b) but DEq is pv; (c) there-
fore EF is . Which was to be demonftrated.
P.
In numbers; let there be DF, 2. DE, ✔ 3. then EF
hall be 2- 3.
PRO P. LXXV.
If from a medial-line DF a D-
medial-line DE commenfurable on-
E-
ly in power to the whole DF, and comprehending with the
whole DF a rational rectangle, be taken away, the remain-
der EF is irrational, and is called a first refidual-line of a
medial.
For EFq (a) to the rectangle FDE. therefore fee-
ing FDE (6) is pv. (c) EF ſhall be j. Which was to be de-
monftrated.
In numbers, let DF be v√ 54, and DE
fore EF is v✔ 54-0 ✓ 24.
PROP. LXXVI.
D
If from a medial-line DF, a
medial-line DE, be taken away
a lem. 26.
10.
b hyp.
C10 & 111
def. 10,
a ſah. 20.
10.
24, there-
b byp.
C 20. and
11 def. 10.
E
F
being commenfurable only in power to the whole DF, and com-
M
prehending
178
The tenth Book of
a byp.
b 16. 10.
C 24. 10.
d cor. 7. 2.
e 27. 10.
a hyp.
prehending together with the whole line DF a medial re&ian-
gle, the remainder EF is irrational, and is called a fecond
refidual of a medial-line.
Becauſe DFq and DEq (a) are pe, (b) therefore
fhall DFqDEq be DEq; c) wherefore DFq+
DEq is pv, Alfo the rectangle FDE, and fo 2 FDF, (a)
is uv, therefore EFq (d) (DFq + DEg- 2 FDE) (e) is
fv. wherefore EF is p. Which was to be demonftrated.
In numbers, let DF be v√ 18. and DE v√ 8. then
EF v √ 18 -v v8.
υν 8ο
A
PROP. LXXVII.
BC
If from a right-line AC be ta-
ken away a right-line AB being in-
commenfurable in power to the whole line AC, and making with
the whole AC that which is composed of their ſquares rational,
and the rectangle contained under them medial, the remain-
der BC is irrational, and is called a Minor-line.
For ACq+ABq (a) it pv. but the rectangle ACB (a) is
bfch. 12. 10. uv; (b) therefore 2 CAB ACq+ABq (c) (2 CAB+
BCq.) (d) therefore ACq+ ABq BCq, (e) therefore
BC is p. Which was to be demonftrated.
In numbers, let AC be y: 18-
€ 7. 2.
d 17. 10.
e 11. def.
10.
a hyp. and
Sch. 12. 10.
b byp.
cfch. 12. 10.
d 7. 2.
e ſch. 12. 10.
and 11. def.
10.
108.
108; AB
18
108. then BC is: 18+√: 108
√ : 18+ √ : 108 — √ : 18 — √
DÈ
PROP. LXXVIII.
If from a right-line DF be
taken away a right-line DE,
being incommenfurable in power to the whole line DF, and with
the whole DF making that which is compofed of their fquares
medial, and the rectangle contained under the fame lines ratio-
nal, the line remaining EF is irrational, and is called a
line making a whole space mediul with a rational space.
For 2 FDE (a) is ev, (b) and DFq + DEq is pv (c)
therefore 2 FDE DFq + DEq (d) (2 FDE -|- EFq)
(e) therefore EF is p. Which was to be demonfirated..
In numbers, let DF be √4 216 +√ 72; DE 216
72. therefore EF is 4 216+ 72—√+216—√72
PROP.
EUCLIDE's Elements.
DE
PROP. LXXIX.
-F
If from a right-line DF be
taken away a right-line DE,
incommensurable in power to the whole DF, and which toge-
ther with the whole makes that which is composed of their
fquares medial, and the rectangle contained under them alſo
medial and incommenfurable to that which is compofed of their
fquares, the remainder is irrational, and is called a line mak-
ing a whole space medial with a medial Space.
For 2 FDE, and FDqDEq (a) are μ- ;
(b) therefore
EFq (c) (DFq + DEq-2 FDE) is p.. (d) and ſo confe-
quently EF is . Which was to be demonftrated.
In numbers; let DF be √ 180 + √ 60. DE ✔✅+
18060, then EF fhall be √ + 180 + √ 60 — √//+
180 — √ 60.
>
4
4
179
a kyp &
24. 10.
b 27 10.
C cor. 2
p11 def. 10
Lemma.
B
M
G
Da
-E
-F
C-
H-
If there be the fame excess between the first magnitu le
BG and the fecond C (MG) as is between the third magni-
tude DF and the fourth H (EF ;) then alternately, the fame
excess fhall be between the first magnitude BG and the third
DF, as is between the ſecond C and the fourth H.
For becauſe that (a) to the equals BM, DE, are added
the equals MG, EF, that is, C, H; the exceſs of the
wholes BG, DF, (6) ſhall be equal to the exceſs of the
parts added C, H. Which was to be demonfirated.
Coroll.
Hence, Four magnitudes Arithmetically proportional,
are alternately alfo Arithmetically proportional.
PRO P. LXXX.
a hyp.
b15.ax. 1.
To an Apotome or re-
A
-BD-C
fidual line AB only one
rational right-line BC, being commenfurable in power only to
the whole AB, is congruent, or can be joyned.
M z
IF
180
The tenth Book of
a 22. 10.
b 22. 10.
c cor. 7. 2.
d lem. 79.10
e hyp. and27.
10.
ffch. 12. 10.
g 27. 10.
a byp.
If it be poffible let fome other line BD be added to
it; (a) then the rectangles ACB, ADB, (6) and fo confe-
quently the doubles of them are ; wherefore fecing
ACq BCq-2 ACB (c) = ABq (0)= ADq+DBq-2
+
ADB, therefore alternately ACq+BCq
BDq (d)
Aq +
— 2 ACB 2 ADB. But AСq + BCq
ADq - BDq (e) it f♥ (f) therefore 2 ACB 2 ADB
Which is abfurd.
ABD
PROP. LXXXI.
-C To a firf medial refidual-line
AB only one medial right-line
BC, being commenfurable only in power to the whole, and
comprehending with the whole line a rational-rectangle,
can be joyned.
Conceive BD to be fuch a line as may be joyned
to it; then becauſe ACq and BCq, as well as ADq and
BDq (a) are μz; (b) alfo ACq -- BCq, and ADq+
b 16. and 24. BDq fhall be us; (c) but the rectangles ACB, ADB, (α)
and fo 2 ACB and 2 ADB are pr; (e) therefore 2 ACB
2 ADB, (f) that is, ACq+BCq ADq - BDq is
10.
c hyp,
d ſch. 12. 10. pv.
e ſch. 27. 10.
f 7.2. and lem
79. 10.
& 27. 10.
ax. I.
(g) Which is abfurd.
PROP. LXXXII, Plate V. Fig 2.
=
Upon a Second medial-refidual-line AB only one medial
right-line BC, commenfurable only in power to the whole,
and with it containing a medial-rectangle, can be joyned.
If it be poffible, let fome other line BD be added to
it; and upon EF make the rectangle EG = ACq
+ BCq; as alfo the rectangle EL = ADq +BDq.
likewife EIABg. Now 2 ACB+ ABq ACq.1-
a 4. 2. and 3. BCqEG; therefore fecing EI ABq, (a) alfo KG
ſhall be 2 ACB, moreover ACq and ECq (b) are uae
(c) therefore EG (ACq- BCq) is μy (d) therefore
the breadth EH is EF. (e) Further, the rectangle
ACB (ƒ) and fo 2 ACB (KG) is pv; (d) therefore KH is
alfo EF. Laftly, becauſe ACq+ BCq (EG) (g)□
2 ACB(KG) and EG: KG:: EH: KH; (k) therefore
EHHK; (4) therefore EK is a refidual-line, whereto
KH is congruent, by the fame reafon alfo fhall KM be
congruent to the faid EK. Which is repugnant to the 80.
prop. of this Book.
b hyp.
C 24. 10.
d 23. 10-
e byp.
f 24. 10.
g
lem. 26.10.
h 1. 6.
k 10. 10.
1 74. 10.
PROP
EUCLIDE's Elements.
181
PROP. LXXXIII.
To a Major-line AB only
one right-line BC can be
A—B-
—
D
C
joyned being incommenfurable in power to the whole, and mak-
ing together with the whole line that which is composed of their
Squares rational, and the rectangle which is contained under
them medial.
Conceive any other BD to be congruent to it; There-
fore whereas. ACq + BCq, and A Dq4 BDq (a) are pa,
their exceſs (2 (6) ACB 2 ADB)(c) is pv. Which is
abfurd; becaufe ACB and ADB are pa by the Hyp.
PROP. LXXXIV.
Unto a line (AB) making
with a rational-space a whole
A-B
·BDC
Space medial only one right-line BC can be joined, being in-
commenfurable in pover to the whole, and making together with
the whole that which is composed of their fquares medial, and
the rectangle which is contained under them rational.
Suppofe fome other BD be congruent alfo to it; (a)
then the rectangles ACB, ADB, (b) and fo 2 ACB and 2
ADB are sa; therefore 2 ACB 2 ADB, (c) that is,
ACqBCq ADq+BDq (d) is pv. Which is abfurd;
fince ACq + BCq, and ADq + BDq are a by the
Hyp.
PRO P. LXXXV. Plate V. Fig. 3.
To a line AB, which with a medial-fpace makes a whole
Space medial, can be joined only one right-line BC, incom-
menfurable in power to the whole, and making with the
whole both that which is composed of their fquares me-
dial, and the rectangle which is contained under them
medial and incommenfurable to that which is composed of
their fquares.
Thoſe things being fuppofed which are done and
fhewn in the 82. prop. of this Book; it is clear that EH
and KH are. EF. Befides, fince, ACq + CBq,
that is, the rectangle EG, (a) is TL ACB, (b) and ſo EG
2 ACB (KG;) and'EG: KG (c): :EH: KH, ſhall EH
be KH; therefore EH is a refidual-line, and the
line congruent to it is KH. In like manner may KM
be fhewn to be congruent to the refidal EK, againſt
the 80. prop. of this Book.
M 3
Third
a 1. hyp,
b lem. 97.
10.
cfch.27.10
d 27. 10.
a hyp.
bſch. 12.
10.
c lem. 79.
10.
d fcb.27.10
a hyp.
b 14. 10.
CI. 6,
182
The tenth Book of
A
Third Definitions.
Rational-line and a refidual being propounded, if
the whole be more in power than the line joined
to the refidual, by the fquare of a right-line commen-
farable unto it in length; then
I. If the whole be commenfurable in length to the
rational-line propounded, it is called a firft refidual-
line.
II. But if the line adjoined be commenfurable in length
to the rational-line propounded, it is called a ſecond re-
fidual-line.
III. If neither the whole nor the line adjoined be
commenfurable in length to the rational-line propounded,
it is called a third refidual-line.
Moreover, if the whole be more in power than the
line adjoined by the fquare of a right-line incom-
menfurable to it in length; then
IV. If the whole be commenfurable in length to the
rational-line propounded, it is called a fourth refidual-
line.
V. But if the line adjoined be commenfurable in length
to the rational-line propounded, it is a fifth refidual.
VI. If neither the whole nor the line adjoined be
commenſurable in length to the rational-line propounded,
it is termed a fixth refidual-line.
PROP. LXXXVI, 87, 88, 89, 90, 91.
A....4 C... 5 B
Ꭰ
E-
G
H-
F
To find out a first, fecond, third,
fourth, fifth, and fixth refidual-line.
Refidual-lines are found out by
fubducting the lefs names or parts
of binomials from the greater.
Ex. gr. Let 6+ √ 20 be a firſt
zo be a firſt reſidual. So that
more concerning the finding
binomial, then ſhall 6 —
it is not neceflary to repeat
of them out,
Lemma,
EUCLIDE's Elements.
183
Lemma. Plate V. Fig. 4, 5·
Let AC be a rectangle contained under the right- lines
AB, AD. Let AD be drawn forth to E, and DE equally
divided in F; and let the rectangle AGE be =
FEq, and
the rectangles AI, DK, FH, finished. Then let the fquare
LMAH be made, and the ſquare NO = GI; and the
lines NSR, OST, produced.
=
=
I fay, . The rectangle AI LM + NO TOq+
SOq, which appears by the conſtr.
2. The rectangle DK LO. For becauſe the rectan-
gle AGE (a) FEq (6) thence are AG, FE, GEO
and fo AH, FI, GI, (a) that is, LM, FI, NO; but
LM, LO, NO (d) are; therefore FI = = (e) LO (f)
= DK = (g) NM.
3. Hence, ACAI
LO NMTR.
DK - · FILM + NO
4. It is manifeft that DF, FE, DE, are.
5. If AE. DE, and AE TV AEq — DEq, (k)
then fall AG, GE, AE be.
6. Also, becauſe AE (1) DE, (m) thence fhall AE, FE,
bea; (n) and ſo AI, FI, that is, LM + NO and LO
area.
7. Becaufe AG*GE, (n) fhall AH, GI, that is, LM,
NO be.
8. But becaufe AE ()
DE, (0) therefore fhall FE,GE
be, (n) and ſo the rectangle FIGI, that is, LO
NO; wherefore feeing LO: NO (p): TS: SO; (q) there-
fore fhall TS, SO be.
9. If AE be put
GË, AE be .
√ AEq DEq, then ſhall AG,
Ic. () Wherefore the rectangles AH, GI, that is, Toq,
50q ball be.
PRO P. XCII. Fig. 4, 5.
If a Space AC be contained under a rational-line AB,
and a first refidual-line AD (AE-DE) the right-line
TS, which containeth the Space AC in power, is a re-
fidual-line.
M 4
Ule
a conftr.
b 17. 6.
C I. 6.
dfch.22.6.
e 9 5.
f 36. 1.
843.1.
h 16. 10.
k 18, and
10. 10.
1 hyp.
m 13. 10.
n 1. 6. and
10. 10.
* before.
O 14. 10.
P 2. 6.
q 10. 10.
г19. 10.&
17. 10.
fi. 6. and
10. 10.
184
The tenth Book of
a hyp.
biz, 10.
C 20. 10.
d lem, 91.
10.
e 74. 10.
a hyp.
b 13. 10.
C 22. 10.
d lem. 74.
10.
e hyp.
f 20. 10.
% 75. 10.
a hyp.
b 22. 10.
C 24. 10.
d 76, 10.
alem. 91. 10.
b byp.
C 20. 10.
d
77. 10.
(a)
Ufe the foregoing Lemma for a preparation to the de-
monftration of this prop. Therefore TS = √ AC,
Alfo AG, GE, AE, are ; therefore fince AE
AB §, (b) alfo AG and GE fhall be
the rectangles AH and GI, that is,
pt. (d) Likewife TO, SO, are
AB, (c) therefore
TOq and SOq are
(e) and confequently
TS is a refidual-line. Which was to be demonftrated.
PROP. XCIII. Plate V.
Fig. 4, 5.
If a Space AC be contained under a rational line AB,
and a fecond refidual AD (AE-DE) the right-line TS,
containing the Space AC in power, is a first medial refidu-
al-line.
Again, by the foregoing Lemma, AG, GE, AE are
; therefore (2) fince AE is AB, (b) alſo AG,
GE, fhall be AB, (c) therefore the rectangles AH,
GI, that is, TOq, SOq are ua; (d) likewife TOSO.
Laftly, becauſe DE(e) AB, (f) the right-angle D',
and the half thereof DK or LO, that is, TOS fhall be
pv; (g) from whence it follows that TS (√ AC) is a firſt
medial refidual. Which was to be domonftrated.
PROP. XCIV. Fig. 4, 5.
If a space AC be contained under a rational-line AB ant
a third refidual AD (AE-DE) the right-line TS containing
in power the Space AC is a fecond medial refidual-line.
µ.,
As in the former, TO and SO are . Therefore be-
caufe DE (a) is AB, (b) the rectangle DI, (c) and fo
DK, or TOS, fhall be ur; therefore TS AC is a
fecond medial refidual. Which was to be demonftrated,
PROP. XCV. Fig. 4, 5.
If a Space AC be contained under a rational-line AB and a
fourth refidual AD (AE-DE) the right-line TS contain-
ing the Space AC in power, is a Minor-line.
As before, TO (a) SO. Therefore becauſe AE
(b) is p AB, c) fhall AI (TOq+SOq) be pv, but, as
before, the rectangle TOS is v; (d) therefore TS = √
AC is a Minor-line. Which was to be demonflrated.
PROP.
EUCLIDE's Elements.
185
PRO P. XCVI. Plate V. Fig. 4 5.
If a space AC be contained under a rational-line AB and
a fifth refidual AD (AE -DE) the right-line TS contain-
ing in power the Space AC, is a line which maketh with a
rational Space the whole pace medial.
For again TO
A
va
SO. Therefore fince AE (a) is
AB (b) alfo AT, that is, TOq+SOq fhall be u.
But, as in 93. the rectangle TOS is pv; (c) whence TS
√ AC is a line which with pv makes a whole ~♥.
Which was to be demonftrated.
PROP. XCVII. Fig. 4, 5.
If a fpace AB contained under a rational-line AB, and a
fixth refidual AD (AE-DE) the right-line TS containing
in power the Space AC is a linc making with a medial rectan-
gle, a whole space mediol.
As often above TO SO. Alfo, as in 96. TOq+
SOq is, but the rectangie TOS is pv, as in 94. (a) Laftly,
TOq-SOqTOS (6) therefore, TSAC is a line
which with u. makęs a whole
V. Which
Which was to be dem.
Lemma. Fig. 6.
=
Upon a right-line DE* apply the rectangles DF ABq,
and DH-ACq, and IK BCq. and let GL be befected in
M, and the line MN drawn parallel to GF.
Then 1. The rectangle DK is = ACq -1- BCq. as the
conftruction manifefts.
=
2. The rectangle ACE=GN or MK. For DK (a) =
ACq+BCq (b) = 2 ACB+ ABq; but ABq (a) DF.
therefore GK (‹) = 2 AÜB; and confequently GN or
MK = ACB.
3. The rectangle DIL = MLq. For becauſe ACq :
ACB (e): : ACB: BCq, that is, DH: MK: : MK :
IK. (e thence is DI: ML:: ML: IL (f) therefore DIL
•
= MLq.
a hyp.
b zz. 10.
c 78. 10.
a lem. 91.
10.
b 79. 10.
*
cor. 16.5.
a confir.
b
7. 2.
C 3. ax. 1.
27.
7. ax. 1.
e 1. 6.
f 17,6.
BC, then DK ſhall be ACq.
—
g 16. 10.
4- If AC be taken
For ACq BCq (DKJ (g) □ ACq.
√
5. Likewife DL DLq- GLq. For becauſe
IK (BCq) (b) thence fhall DI be
DH (ACq)
(k) therefore✔ DLq — GLq □ DL.
IL;
6. Alfo DL GL. For ACq+BCqπ (1) 2 ACB.
That is, DKGK; (m) therefore DL
GL.
7. Belt
h 10. 10.
k 18. 10.
llem.26.10
m 10: 10.
$6
The tenth Book of
a byp.
b lem. 97.
10.
c ſch. 12.
10.
d 21. 1c..
e 22. and
24. 10.
f
£ 23. 10.
g 13. 10.
h fch. 12.
10.
k 74. 10.
1idef. 85.10.
m lem. 97.
10.
a hyp.
b. lem. 97.
10.
€ 24. 10.
d 23.10.
← hyp. and
fch. 12. 10.
f21. 10.
g 13. 10.
hfcb. 12.
10.
k 74. 10.
I lem. 97.
10.
m 2. def.
85. 10.
a 23. 10.
blem. 26.
JO.
c 1. 6. and
10. 10.
d ſch. 12. 10.
€74 10.
£ 3. def. 85.
10.
g lem. 97.
10,
7. But if AC be taken
√ DLq
GLq.
BC, then DL ſhall be
PROP. XCVIII. Plate V. Fig. 6.
The Square of a refidual-line AB (AC-BC) applyed to a
rational-line DE, makes the breadth DG a first residual-line.
Do as is enjoined in the Lemma next preceding. Then
becauſe AC, BC, (a) are ; (b) alfo DK (ACq+
BCq) fhall be ACq. (c) Therefore DK is pv; (d)
wherefore DL is DE. (e) Likewife the rectangle
GK (2 ACB) is pr; (f) therefore GL is
and confequently DL GL. (b) But DLq
<
DE, (g)
GLq,
(k therefore DG is a refidual, () and that of the firft
order (becauſe (m) AC BC, and therefore DL
√ DLqGLq.) Which was to be demonftrated.
PROP. XCIX. Fig. 6.
The fquare of a firſt medial refidual-line AB (AC-BC)
applied to a rational-line DE, makes the breadth DG a fecond
refidual-line.
Suppofing the foregoing Lemma; becaufe AC and BC
(a) are u, (b) thence hall DK (ACq+BCq) be
ACq; (c) wherefore DK is uv; (d) therefore DL is p
DE; (e) alfo GK (2 ACB) is §; (f) therefore GL is p
E; (g) wherefore DL GL. (b) But DLq
GLq; (k) therefore DG is a refidual-line. And becauſe
DL is TL/DLq — GLq, (m) therefore ſhall DG be
a fecond refidual. Which was to be demonftrated.
PROP. C. Fig. 6.
The fquare of a fecond medial refidual-line AB (AC-BC)
applied to a rational-line DE, makes the breadth DG a third
refidual-line.
DE; alfo
DE; (b) likewife
Again DK is uv. (a) wherefore DL is
GK is pv, (a) whence GL is
DKGK; (c) wherefore DL
GL; (d) but DLq
GLq. (e) therefore DG is a refidual-line, and that
of (f) the third order, (g) becauſe
GLq. Which was to be demonfirated.
DL√ DLq.
PROP.
{
EUCLIDE's Elements.
x87
PROP. CI. Plate V. Fig. 6.
<
DE. (c)
The Square of a Minor-line AB (AC-BC) applied to
a rational-line DE, makes the breadth DG a fourth refidual.
As before, ACq-+-BCq, that is DK, is pv. (a) there-
fore DL is a DE. but the rectangle ACB, and fo
GK (2 ACB) * is uv. (b) wherefore GL is
therefore DL GL. (d) but DLq GLq, and be-
caufe * ACq BCq, (e) thence hall DL be
DLq - GLq. (f) therefore DG has the conditions re-
quired to a fourth refidual, Which was to be demon-
frated.
PROP. CII. Fig. 6.
√
The fquare of a line AB (AC-BC) which makes with
à rational ſpace the whole space medial, applied to a rational-
line DE, makes the breadth DG a fifth refidual-line.
For, as above, DK is ur. (a) wherefore DL is p
DE alfo GK is pv. (b) whence GL is DE. (e) there-
fore DL GL. (d) but DLq GLq. Moreover
DL (e) ✓ DLq-GLq. wherefore DG (f) is a fifth
refidual. Which was to be demonflrated,
PROP. CIII. Fig, 6.
The fquare of a line AB (AC — BC) making with a
medial Space the whole space medial, applied to a rational-
line DE, makes the breadth DG a fixth refidual-line.
As above, DK and GK are e; (a) wherefore DL
and GL are . DE. alfo DK (4) GK. (c) whence
DL GL. (d) therefore DG is a refidual. (6) And
whereas ACqBCq. and fo DL √ DLq - GLq,
(e) therefore DG fhall be a fixth refidual. Which was
to be demonftrated,
PROP. CIV.
"A right-line DE commen- A-
furable in length to a refidual
AB (AC-BC) is it felf alſo a D-
rejidual, and of the fame order.
B
- IE
E
C
a 21. 10.
*byp.
b 23. 10.
C 13. 10.
d ſch. 12.
10.
e lem. 97.
10.
£4.def.85.
10.
2 23. 10.
b 21. 10.
C 13. 10.
d ſch, 12.
10.
e lem. 97•
10.
£5. def.85.
10.
a 23. 10.
b hyp, and
lem, 97.10.
C 10. 10.
d 74. 10.
e6.def.85,
10.
Lemma.
Let AB: DE:: AC: DF, and AB DE.
I
188
The tenth Bock of
a lem. 66.
10.
b 10. 10.
a 12. 6.
b lem. 103.
10
chypo
d 67. 10.
e by def. 85.
10.
a 12. 6.
b lem. 103.
IO.
:
I fay AC+ BCDF+ EF. For AC: BC (a): :
DF: EF. therefore by compounding AC-|-BC: BC:
DFEF: EF therefore by permutation AC + BC :
DF--EF:: BC: EF. (a) but BC EF; (b) there-
fore AC+BC DFTEF, Which was to be dem.
(a) Make AB: DE:: AC: DF, (b) therefore AC+
BCDF + EF. therefore feeing AC + BC (c) is a
binomial, (d) DF EF fhall be a binomial too, and of
the fame order. (e) wherefore DF EF is a refidual of
the fame order with AC-BC. Which was to be de-
monflrated.
A
B
1
D E
F
C
PROP. CV.
A right-line DE commenfurable
to a medial refidual-line AB (AC
BC) is it felf a medial refiduul,
and of the fame order.
Again (a) make AB: DE:: AC: DF: (b) whence AC
+ BC DF + EF. (c) therefore DF EF is a bime-
dial of the fame order with AC+ BC, (d) and confe-
quently DF-EF fhall be a medial reñidual of the
fame order with AC- BC.
c.68. 10.
d 75. and
76. 10.
Which was to be dem.
PROP. CVI.
A
B
C
commen--
1
a em. 103.
10.
D
I
E
F
A right-line DE
furable to a Minor - line AB
(AC- BC) is it felf alfo a
Minor-line.
Make AB : DE :: AC : DF. (a) then is AC + BC
LFEF. but AC+BC (6) is a Major line; (c)
therefore DF+ EF is alfo a Major line; (d) and confe-
EF is a Minor line. Which was to be dem.
quently DF
b hyp.
c 69. 10.
d
77. IO.
A
B C
D
1-
Ε
F
PROP. CVII.
A right-line DE commenfurable
to a line AB (AC- BC) which
makes with a rational space the
whole space medial, is it ſelf alſo a
line making with a rational ſpace the whole fpace medial.
For, accordingly as in the former, we may fhew
EF to contain in power pv and pv. (a) whence
EF is a line making, &c.
a 78. 10.
DF
DF
PROP.
EUCLID E's Elements.
189
PROP.
CVIII.
A right-line DE commenfurable A
to a line AB (AC~ BC) which
with a medial Space makes the
whole space medial, is it ſelf a line D
making with a medial space the
whole space medial.
For according to the preceding DF
B C
1.
-1
E
F
EF fhall con-
tain in power 2 a. (a) therefore DF-EF fhall be, as
in the Prop.
PROP CIX. Plate V. Fig. 7.
A medial rectangle B being taken from a rational recan-
gle A+B, the right-line H which containeth in power the
Space remaining A, is one of theſe tavo irrational-lines、 viz.
either a refidual-line, or a Minor-line.
Upon CD make the rectangles CIA + B, and
FI B. whence CE (a) = A≈Hq. wherefore becauſe CI
(b) is fv. (c) therefore CK is CD; but becaufe FI (b)
is v, (d fhall FK be ' CD; (e) whence CK FK;
(f) therefore CF is a refidual-line. Wherefore if CK
be✓CKq - FKq. (g) then CF fhall be a first
reſidual; (4) therefore CE (H) is a refidual-line. But
if CK √ CKq—FKq (4) then CF fhall be a fifth
refidual; and confequently H(CE) (1) fhall be a
Minor-line. Which was to be demonſtrated,
PROP. CX. Fig. 7.
:
A rational rectangle B being taken away from a medial
redlangle AB, other two irrational-lines are made, name-
ly; either a firft medial refidual-line, or a line making with
a rational ſpace the whole space medial.
Upon CD the propounded make the rectangle CI
AB, and FIB (a) whence CE
B (a) whence CE A Hq.
Therefore becauſe CI (6) is uv; (c, fhall CK be CD.
§ a
but becauſe FI (b) is fr (d) thence FK CD. (e)
whence CKFK. (f) therefore CF is a refidual, (g)
and that a fecond. If CK CKq-FKq, (b) then
H (✓ CE) is a firſt medial refidual. But if CK /
CKq- FKq, (4) then fhall CF be a fifth refidual; and
(1) confeqeently Н (√ CE) fhall be a line making
with pr. Which was to be demonftrated.
I
a 79. 10.
a 3. ax. 1.
b hyp. and
confir.
C 21. 10.
d 23. 10.
e 13. 10.
f
74. 10.
g1.def. 85.
10.
h 92. 10.
k 4.def.85.
10.
195. 10.
a 3. ax. 1.
b hyp. and
conftr.
C 23. 10.
d 21. 10.
e 13. 10.
f 74. 10.
g2.def.85.
10.
h 93. 10.
k 5.def.85.
10.
PROP. 196. 10.
190
The tenth Book of
a 3. ax. 1.
b 23. 10.
c hyp.
d 10. 10.
e 74. 10.
PRO P. CXI. Plate V. Fig. 7.
A medial Space B being taken away from a medial ſpace
A--B, which is incommenfurable to the whole A+B,
the other two irrational-lines are made, viz. either a fecond
medial refidual-line, or a line making with a medial ſpace
the whole ſpace mcdial.
C
ρ
Upon CD make the rectangles CIA+B, and
FIB. (a) wherefore CEÅ Hq. Becauſe there-
fore CI is, (b) thence CK is p' CD. and in like
manner FK CD. Likewife becauſe CI (c) FI,
(d) therefore CK FK (e) wherefore CF is a refidual,
(f) namely a third, If CK / Ckq-Fkq, (g) whence
H (CE) fhall be a fecond medial refidual ; but if
h 6. def. 85. CK CKq-FKq. (b) then fhall CF be a fixth re-
/
fidual; (4) wherefore A fhall be a line making uv with
Which was to be demonftrated.
£ 3. def, 85.
10.
g 94. 10.
10.
k
97. 10.
a 98. 10.
b 74. 10.
c 1, def. 85.
10.
d 37. 10.
e 1. def. 48.
10.
f 12, 10.
g cor. 16.
10.
h ſch. 12.
10.
PRO P. CXII. Fig. 8.
A refidual-line A is not the fame with a binomial-line.
Upon BC propounded & make the rectangle CD=Aq,
Therefore feeing A is a refidual, (a) BD fhall be a firſt
refidual, to which let DE be the line congruent or that
may be adjoined; (b) wherefore BE, DE, are ; (c)
and BE BC. If you conceive A to be a binomial,
then BD is a firft binomial, whofe names let be BF, FD;
and let BF be FD; (d) therefore BF, FD are
and BF (e) BC. therefore fince ECBE, (ƒ)
fhall BE be BF. (g) and thence BE FE, (b)
therefore, FE is p. Likewife becaufe BE DE, (k) ſhall
FE be DE; (1) wherefore FD is a refidual, and fo
FD is p. but it was fhewn . which are repugnant;
therefore A is falfely conceived to be a binomial.
Which was to be demonftrated.
k 14. 10.
1 74. 10.
The
EUCLIDE's Elements.
191
The names of the 13 irrational-lines differing
one from another.
1. A Medial-line.
2. A binomial-line; of which there are fix fpecies.
3. A first bimedial-line.
4. A fecond bimedial.
5. A Major-line.
6. A line containing in power a rational fuperfi-
cies, and a medial fuperficies.
7. A line containing in power two medial fuperfi-
cies.
8. A refidual-line; of which there are alfo fix kinds.
A firft medial refidual-line.
9.
10.
A fecond medial refidual-line.
11. A Minor-line.
12. A line making with a rational fuperficies the
whole fuperficies medial.
13. A line making with a medial fuperficies the
whole fuperficies medial.
Since the differences of breadths argue differences of
right-lines, whofe fquares are applied to feme rational-line,
and it is demonftrated in the preced. Propofitions that the
breadths which ariſe from applying of the ſquares of theſe
13 lines to differ one from another, it evidently feilorus that
theſe 13 lines do alfo differ one from another.
I
PROP. CXIII. Plate V. Fig. 8.
The Square of a rational-line ▲ applied to a binomial
BC (BD+ DC) makes the breadth EC a refidual-line,
whofe names EH, CH, are commenfurable to the names BD,
JC, of the binomial-line, and in the fame proportion (EH :
BD:: CH: DC;) and moreover, the refidual-line EC
which is made, is of the fame order with BC the binomial.
Upon LC the lefs name (a) make the rectangle DF
Aq=BE, whence BC: CD (6) : : FC : CE. there-
fore by divifion, BD: DC :: FE: EC. And whereas
BD (c) DC, (d) thence FE fhall be EC; Take EG
Then EH,
EC, and make FG: GE:: EC: CH.
and CH fhall be the names of the refidual EC, where-
unto all is agreeable that is propounded in the theorem.
For by compounding, FE : GE (EC) :: EH: CH. there-
fore FH: EH():: EH: CH (f): FE: EC():: BD:
DC. wherefore fince BD (g) DC (b) thence shall EH
a cor. 16.6.
b 14. 6.
c byp.
d 14. 5.
e 12. 5.
f before.
g hyp.
be
h 10. 10.
192
The tenth Book of
1 16. 10.
m 21. 10.
n fcb. 12.
10.
0 74. 10.
p 10. 10.
915. 10.
r 12. 10.
f 1. def. 48.
be EH, (b) and FHq EHq. Therefore becaufe
k cor. 20 6. FHq: Ehiq (k): FH: CH:/) fhall FH be CH; (/)
and fo FC CH. Moreover CD (g) is , and DF (Aq)
(g) is fv; (m) therefore FC is §, ČD, whence alfo CH
is CD; (2) therefore EH, CH are & and, as be-
p
fore; (0) therefore EC is a refidual-line, to which Hmay
be joined. Furthermore EH: CH (ƒ) : : BD : DC, and
fo by permutation EH: BD :: CH´: DC; whence be-
caufe CH (ƒ) DC, (p) fhall EH be BD. But
fuppofe BD√ BUq — TCy, (9) then fhall EH be
EHq. CHq. Alfo if BD propounded,
then ſhall EH be to the fame p. (/) that is, if BC be
a firſt binomial, (t) EC fhall be a first refidual. In like
manner, if DC be to the propounded §, (~) then is
CH to the fame . (a) that is, if BC be a fecond
binomial, (x) EC fhall be a fecond refidual: And if
this be a third binomial, then that fhall be a third re-
fidual, &c. But if BD be y BDq — DCq, (y) then
fhall EH be √ EHq - CHq. therefore if BC be a
4, 5, or 6 binomial, EG fhall be likewiſe a 4, 5, or 6,
refidual. Which was to be demonftrated.
10.
t 1. def. 85.
10.
u 2. def. 48.
10.
x 2. def. 85.
10.
y 15. 10.
a cor. 16. 6.
b 12.6.
€ 14. 6.
d 19. 5.
e hyp.
f 10. 10.
g cor. 20. 6.
h 10. 10.
k cor. 16. 10.
1 21. 20.
m 12. 10.
ných. 12. 10.
0 37. 10.
P 10. 10.
PROP. CXIV. Plate V. Fig. 9.
The ſquare of a rational line A applied to a refidual-line
BC (BD — CD) makes the breadth BE a binomial; whofe
names BE, GE are commenfurable to the names BD, BC of
the refidual-line LC, and in the fame proportion, and more
over, the binomial-line which is made (BE) is of the fame
order with the refidual-line (BC.)
(a) Make the rectangle DF= Aq. and BF: FE (6)
:: EG: GF; whence for that DF=Aq=CE, (c) there-
fore BD BC: BE: BF. therefore by converfion of pro-
portion BD: CD :: BE: FE :: EG: GF :: (d) GB: EG.
but BD (e) CD; (f) therefore BGGE. therefore
becauſe BGq: GEq(g): : BG: GF. (b) ſhall BG be TGF;
(k) and fo BG BF. moreover BD (e) is p, and the rec-
tangle DF (Aq) (e) is fv. (1) therefore BF is p□ BD. (m)
therefore alfo BG is BD. (n) therefore BG, GE
are. (o) wherefore BE is a binomial. Laftly, be-
caufe BD: CD:: BG: GE. and by permutation BD : BG
:.: CD : GE, and BD BG (p) thence fhall CD be
GE. therefore if CB be a firft refidual, BE fhall be a firſt
binomial, &c. as in the prec. therefore, &c.
PROP.
EUCLIDE's Elements.
193
PRO P. CXV. Plate V. Fig. 10.
If a Space AB be contained under a refidual-line AC
(CEAE) and a binomial CB, whoſe names CD, db,
are commenfurable to the names CE, AE, of the refidual-
line, and in the fame proportion (CE: AE::CD : DB.)
then the right-line F which containeth in power that Space
AB, is rational.
Let G be . and make the rectangle CH-Gq; (4)
then ſhall BH (HI-IB) be a refidual-line, and Hl (a)
п CD (b) п CE. (a) and BI DB. (a) and HI:
BI:: CD: DB (b): CE: EA. therefore by permutation
HI: CE :: BI: EA. (c) therefore BH: AC:: HI : CE
:: BI: EA. wherefore fince (d) HICE, (e) thence
BH AC. (f) therefore the rectangle HC BA.
But HC (Gq) (b) is pv. (g) therefor BA (Fq) is pr. and
confequently F is §. Which was to be demonftrated.
Coroll.
a 113. 10.
b hyp.
c 19. 5.
d 12.10.
e 10. 10.
f 1. 6. and
10. 10.
g ſch. 12.
10.
Hereby it appears that a rational fuperficies may be
contained under two irrational right-lines.
PROP. CXVI. Fig. 11.
Of a medial-line AB are produced infinite irrational-lines
BE, EF, &c. whereof none is of the fame kind with any of
the precedent.
Let AC be propounded f. and AD a rectangle con-
tained under AC, AB; (a) therefore AD is v. Take
BE√ AD. (b) then BE is p, and the fame with none
of the former. For no fquare of any of the former being
applied to p, makes the breadth medial. Let the rec-
tangle DE be finiſhed, (a) then DE fhall be py, and (b)
confequently EF (V DE) fhall be p, and not the fame
with any of the former, for no fquare of the former
being applied to , makes the latitude BE; there-
fore, &c.
a iem. 38.
10.
bil
b 11. 10.
N
PROP.
194
The tenth Book of
2 47.1.
€ 9. 10.
PRO P. CXVII. Plate V. Fig. 12.
Let it be required to shew that in fquare figures BD, the
diameter AC is incommenfurable in length to the fide AB.
For ACq: ABq (a) :: 2 : 1 (b)
1 (b): : not Q: Q (c)
b cor. 24. 8. therefore AC AB. Which was to be demonftrated.
This Theorem was of great note with the ancient Phi-
lofophers; fo that he who did not underſtand it was
efteemed by Plato undeferving the name of a man, but
rather to be reckoned among brutes.
The End of the Tenth Book.
THE
[ 195 ]
The ELEVENTH BOOK
O F
EUCLIDE's
ELEMENT S.
I.
A
Definitions.
Solid is that which hath length, breadth
and thickness.
II. The term, or extreme of a folid is a
Superficies.
III. A right-line AB, Plate V. Fig. 13. is perpendi-
cular to a Plane CD, when it makes right-angles ABD,
ABE, ABF, with all the right-lines BD, BE, BF, that
touch it, and are drawn in the faid Plane.
IV. A Plane AB, Fig. 14. is perpendicular to a Plane
CD, when the right-lines FG, HK, drawn in one Plane
AB to the line of common fection of the two Planes
EB, and making right-angles therewith, do alfo make
right-angles with the other Plane CD.
V. The inclination of a right-line AB, Fig. 15. to a
Plane CD, is when a perpendicular AE is drawn from
A the higheſt point of that line AB to the plane CD,
and another line EB drawn from the point E, which
the perpendicular AE makes in the plane CD, to the
end B of the faid line AB which is in the fame plane,
whereby the angle ABE which is contained under the
infifting-line AB, and the line drawn in the plane EB
is acute.
VI. The inclination of a plane AB, Fig. 16. tô a
plane C, is an acute angle FGH contained under the
right-lines FH, GH which being drawn in either of the
planes AB, CD to the fame point H of the common
fection BE, make right-angles FHB, GHB, with the
Common fection BE:
Na
Wit
196,
The eleventh Book of
VII. Planes are faid to be inclined to other planss
in the fame manner, when the faid angles of inclina-
tion are equal one to another.
VIII. Parallel planes are thoſe which being pro-
longed never meet.
IX. Like folid figures are fuch as are contained un-
der like planes equal in number.
X. Equal and like folid figures are fuch as are con-
tained under like planes equal both in multitude and
magnitude.
XI. A folid angle is the inclination of more than
two right-lines which touch one another, and are not
in the fame fuperficies.
Or thus
A folid angle is that which is contained under more
than two plane angles not being in the fame fuperficies,
but confifting all at one point.
XII. A Pyramid is a folid figure comprehended
under divers planes fet upon one plane (which is the
bafe of the pyramid,) and gathered together to one
point.
XIII. A Priſm is a folid figure contained under
planes, whereof the two oppofite are equal, like, and
parallel; but the others are parallelograms.
XIV. A Sphere is a folid figure made when the di-
ameter of a femicircle abiding unmoved, the femicircle
is turned round about, till it return to the fame place
from whence it began to be moved.
Coroll.
Hence, all the rays drawn from the center to the
fuperficies of a fphere, are equal amongſt themſelves.
XV. The Axis of a fphere, is that fixed right-line,
about which the femicircle is moved.
XVI. The Center of a ſphere, is the fame point
with the center of the femicircle.
XVII. The Diameter of a fphere, is a right-line
drawn thro' the center, and terminated on either fide
in the fuperficies of the fphere.
XVIII. A Cone is a figure made, when one fide of
a rectangled triangle (viz. one of thoſe that contain
the right angle) remaining fixed, the triangle is turn-
ed round about till it return to the place from whence
it
EUCLIDE's Elements.
197
it first moved. And if the fixed right-line be equal to
the other which containeth the right-angle, then the
Cone is a rectangled Cone: But if it be lefs, it is an
obtufe-angled Cone; if greater, an acute-angled Cone.
XIX. The Axis of a Cone is that fix'd line about
which the triangle is moved.
XX. The Bafe of a Cone is the circle, which is de-
fcribed by the right-line moved about.
XXI. A Cylinder is a figure made by the moving
round of a right-angled parallelogram, one of the
fides thereof, (namely, which contain the right-angle)
abiding fix'd, till the parallelogram be turned about
to the fame place, where it began to move.
XXII. The Axis of a Cylinder is that quiefcent right--
line, about which the parallelogram is turned.
XXIII. And the Baſes of a Cylinder are the circles
which are defcribed by the two oppofite fides in their
motion.
XXIV. Like Cones and Cylinders, are thoſe both
whofe Axes and Diameters of their Bafes are propor-
tional.
XXV. A Cube is a folid figure contained under fix
equal fquares.
XXVI. A Tetraedron is a folid figure contained
under four equal and equilateral triangles.
XXVII. An Octaedron is a folid figure contained
under eight equal and equilateral triangles.
XXVIII. Â Dodecaedron is a folid figure contain-
ed under twelve equal, equilateral, and equiangular
Pentagones.
XXIX. An Icofaedron is a folid figure contained
under twenty equal and equilateral triangles.
XXX. A Parallelepipedon is a folid figure contained
under fix quadrilateral figures, whereof thoſe which
are oppofite are parallel.
XXXI, A folid figure is faid to be infcribed in a folid
figure, when all the angles of the figure infcribed are
comprehended either within the angles, or in the fides,
or in the planes of the figure wherein it is infcribed.
XXXII. Likewife a folid figure is then faid to be
circumfcribed about a folid figure, when either the an-
gles, or fides, or planes of the circumfcribed figure
touch all the angles of the figure which it contains.
N 3
PROP.
198
The eleventh Book of
4
2 10. ax. I.
PROPOSITION I. Plate V. Fig. 17.
One part AC of a right-line cannot be in a plane fuper
ficies, and another part of it CB above the fame.
Produce AC in the plane directly to F. If you con-
ceive CB to be drawn ftrait from AC, then two right-
lines AB, AF, have one common fegment AC. (a) Which
is impoffible.
PROP. II. Fig. 18.
If tavo right-lines AB, CD, cut one another, they are in
the fame plane: And every triangle DEE is in one and the
fame plane.
For imagine EFG, part of the triangle DEB, to be in
one plane, and the part FDGB to be in another, then
EF part of the right-line ED is in a plane, and the other
a 10. ax. 1. part elevated upwards. (a) Which is abfurd. Therefore
the triangle EDB is in one and the fame plane; and fo
alfo are the right-lines ED, EB; (a) wherefore the
whole lines AB, DC, are in one plane. Which was te
be demonftrated.
PROP. III. Fig. 19.
If two planes AB, CD, cut one the other, their common
fection EF is a right-line.
If EF the common fection be not a right-line, (a)
a 1. pest. I then in the plane AB draw the right-line EGF; (a) and
in the plane CD the right-line EHF. therefore two right-
lines EGF, EHF include a fuperficies. (b) Which is
abfurd.
b
14. ax. I.
a conftr.
b 15. 1.
C 4. I.
dſ:h. 34, 1.
e 29. 1.
PROP. IV. Fig. 20.
If at E the common fection of two right-lines AB, CD,
a right-line EF stands at right-angles to them, it shall also
be at right-angles to the plane ACBD drawu thro' the faid
lines.
Take EA, EC, EB, ED, equal one to the other, and
join the right-lines AC, CB, BD, AD draw any right-
line GH thro' E, and join FA, FC, FD, FB, FG, FH.
Becauſe AE is (a) EB, and DE (a) = EC, and the
ans le AED (6) CEB () therefore AD is
CB, (c)
likewife AC DB; (d) therefore AD is parallel to CB,
(d) and AC to BD. (e) wherefore the angle GAE=
EBH
1
EUCLIDE's Elements.
199
EBH, and the angle AGE EHB.
EHB. But alſo AE (ƒ)
EB; (g) therefore GE=EH, (g) and AG=BÍ.
whence by reafon of the right-angles, by the hyp. and
ſo equal, at E, (b) the baſes FA, FC, FB, FD, are equal.
Therefore the triangles ADF, FBC, are equilateral one
to another, (k) and thence the angle DAF = BCF.
Therefore in the triangles AGF, FBH, the fides FG,
FH() are equal; and fo by confequence the triangles
FEG and FEH are mutually equilateral; (m) therefore
the angles FEG, FEH are equal, and (2) fo right-angles.
In like manner, FE makes right-angles with all the lines
drawn thro' E in the plane AD, BC, (0) and is there-
fore perpendicular to the faid plane.
PROP. V. Plate V. Fig. 21.
If a right-line AB be erected perpendicular to three right-
lines AC, AD, AE, touching one the other at the common
fection, those three lincs are in the fame plane.
For AC, AD, (a) are in one plane FC; (a) and AD,
AE, are in one plane BE, which if you conceive to be
feveral planes, then let their interfection (b) be the right-
line AG; therefore becauſe BA by the Hyp. is perpen-
dicular to the right-lines AC, AD, (c) and fo to the plane
FC, (d) it is alſo perpendicular to the right-line AG.
therefore (fince (a) that AB is in the fame plane with
AG, AE) the angles BAG, BAE, are right-angles, and
confequently equal, the part and the whole. Which is
abfurd.
PRO P. VI. Fig. 22.
If two right-lines AB, DC, be created perpendicular to
one and the fame plane EF, thofe right-lines AB, DC, are
parallel one to the other.
Draw AD, whereunto let DG=AB be perpendicular
in the plane EF, and join BD, BG, AG. Becaufe in
the triangles BAD, ADG, the angles DAB, ADG (a) are
right-angles, and AB (b) DG, and AD is common, (c)
therefore BDAG. whence in the triangles AGB,
BGD, equilateral one to the other, the angle BAG is
(d) BDG; of which fince BAG is a right-angle,
BDG fhall be fo alfo, but the angle GDC is fuppofed
right, therefore the right-line GD is perpendicular to
the three lines DA, DB, CD, (e) which are therefore in
the fame plane (ƒ) wherein AB is. Wherefore fince AB
N +
and
f confir.
g 26. 1.
h 4. 1.
k 8. 1.
1 4. I.
m 8.1.
n 10. def. I
03.def. 11.
a 2. 11.
b 3 11.
C 4. 11.
d 3. def. 11
a hyp.
b confir
C 4.1
d S. I.
•
e 5. I.
f z t
200
The eleventh Book of
g 28. I.
23.11.
and CD are in the fame plane, and the internal angles
BAD, CDA, are right-angles, (g) AB and CD fhall be
parallels. Which was to be demonftrated.
PRO P. VII. Plate V. Fig. 23.
If there are two parallel right-lines AB, CD, and any
points E, F, be taken in both of them, the line EF which is
joined at theſe points, is in the fame plane, with the paral-
lels AB, CD.
Let the plane in which AB, CD, are, be cut by ano-
ther plane at the points E, F; then if EF is not in the
plane ABCD, it ſhall not be the common fection. There-
fore let EGF be the common fection; which (a) then is
a right-line, therefore two right-lines EF, EGF, in-
b 14. ax. 1. clude a fuperficies. (b) Which is abſurd.
24. II.
b 7. 11.
c 3. def. 11.
d 29. 1.
€ 4. 11.
PROP. VIII. Fig. 24.
If there are two parallel right-lines AB, CD. whereof
one AB, is perpendicular to a plane EF, then the otber CĎ
fhall be perpendicular to the fame plane EF.
The preparation and demonftration of the fixth of this
Book being transferr'd hither; the angles GDA, and
GDB are right-angles: (a) Therefore GD is perpendi-
cular to the plane, wherein are AD, DB, (b) (in which
alfo AB, CD, are.) (c) therefore GD is perpendicular to
CD; but the angle CDA is alſo (d) a right-angle, (e)
therefore CD is perpendicular to the plane E F
Which was to be demonftrated.
PROP. IX. Fig. 25.
Right-lines (AB, CD) which are parallel to the fame
right-line EF, but not in the fame plane with it, are alſo
parallel one to the other.
In the plane of the parallels AB, FF, draw HG per-
pendicular to EF; aifo in the plane of the parallels EF,
CD, draw IG perpendicular to EF. (a) Therefore EG is
perpendicular to the plane wherein HG, GI are; and
AH, CI are perpendicular to the fame plane, (c) there-
fore AH and Cl are parallels. Which was to be demon.
PROP.
EUCLIDE's Elements.
201
PROP. X. Plate V. Fig. 26.
If ewo right-lines AB, AC, touching one another be pa-
rallel to two other right-lines ED, DF, touching one ano-
ther, and not being in the fame plane, thofe right-lines
contain equal angles, BAC, EDF.
Let AB, AC, DE, DF, be equal one to the other,
and draw AD, BC, EF, BE, CF. Since AB, DE, (a)
are parallels and equal, (b) alfo BE, AD, are parallels
and equal. In like manner CF, AD, are parallel and
equal; (c) therefore alſo BE, FC, are parallel and equal.
(d) Therefore BC, EF are equal. Wherefore fince the
triangles BAC, EDF, are of equal fides one to the other,
the angles BAC, EDF (e) ſhall be equal. Which was to
be demonflrated.
PROP. XI. Fig. 27.
From a point given on high A to draw a right-line AI
perpedicular to a plane below BC.
In the plane BC draw any line DE; to which from
the point A (a) draw the perpendicular AF, to the fame
DE through F in the plane BC (6) draw the perpendicu-
lar FH, then to FH (a) draw the perpendicular AI, this
fhall be perpendicular to the plane BC.
For thro' I (c)let KIL be drawn parallel to DE. Be-
caufe DE (d) is perpendicular to AF, and FH. (e) there-
fore DE fhall be perpendicular to the plane IFA; and ſo
alfo KL (f) is perpendicular to the fame plane; (g) there-
fore the angle KIA is a right-angle, but the angle AIF
is alſo (b) a right-angle; () therefore AI is perpendicular
to the plane BC. Which was to be done.
PROP. XII. Fig. 28.
In a plane given BC, at a point given therein A, to erect
• perpendicular line AF.
From fome point D without the plane, (a) draw DE
perpendicular to the faid plane BC, and joining the points
A, E, by a line AE, (b) draw AF parallel to DE (c) it
is apparent that AF is perpendicular to the plane BC,
Which was to be done.
This and the preceding problem are practically per-
formed by applying two Squares to the point given; as
appears by 4. 11.
PROF.
a hyp. ànd
conſtr.
b 33. 1.
C 2. ax. I.
and 9. 11.
d 33. I.
e 8. I.
a 12. I.
b II I.
C 31. I.
d confir.
e 4. II.
f 8. 11.
g 3.def. 11.
h confir.
1
4. II.
a 11. II.
b 31. 1.
C 8.11.
202
The eleventh Book of
a 6. 11.
hyp. and
alef. 11.
b 17. 1.
a 11. 11.
b31
31. 1.
€ 9. II.
PROP. XIII. Plate V. Fig 29.
At a point given C in a plane given AB, tavo right-lines
CD, CE, cannot be erected perpendicular to the faid given
plane on the fame fide.
For both CD, and CE (a) fhould then be perpendicu-
lar to the plane AB, and confequently parallel; which
is repugnant to the definition of parallel-lines.
PROP. XIV. Fig. 30.
Planes CD, FE, to which the fame right-line AB is per-
pendicular, are parallel.
If you deny this; then let the planes CD, EF meet,
fo that their common fection be the right-line GH, in
which take any point I, draw to it the right-lines IA,
IB, in the faid planes, whereby in the triangle IAB, two
angles IAB, IBA (a) are right-angles. (b) Which is abfurd.
PRO P. XV. Fig. 31.
If two right-lines AB, AC, touching one the other, are
parallel to two other right-lines DE, DF, touching one the
other, and not being in the fame plane with them, the planes
BAC, EDF, drawn by thofe right-lines are parallel one to
the other.
From A (a) draw AG perpendicular to the plane EF.
(b) and let GH, GI be parallel to DE, DF. (c) theſe alfo
fhall be parallel to AB, AC. Therefore fince the an-
d 3. def. 11. gles, IGA, HGA, (d) are right-angles, alſo CAG, BAG,
(e) fhall be right-angles; (f) therefore GA is perpendi-
cular to the plane BC; but the fame is perpendicular
to the plane EF, (b) therefore the planes BC, EF, are
parallel. Which was to be demonfirated.
e 29. 1.
4.11.
f
g conſtr.
h 14. 11.
87
11.
PROP. XVI. Fig. 32.
If two parallel planes AB, CD, are cut by fome other
plane HEIGF, their common fections EH, GF are parallel
one to the other.
For if they are faid to be not parallel, then, fince they
are in the fame cutting plane, they muſt meet fome
where, fuppofe in I, wherefore fince the whole lines
HEI, FG1 (a) are in the planes AB, CD, produced, the
planes alfo fall mnect. contrary to the Hyp.
t
PROP
1
EUCLIDE's Elements.
203
PROP
XVII. Plate V. Fig. 33.
If two right-lines ALB, CMD, are cut by parallel planes
EF, GH, IK: they shall be cut proportionally, (AL: LB
:: CM: MD.)
Let the right-lines AC, BD, be drawn in the planes EF,
IK; as alſo AD meeting the plane GH in the point N,
and join NL, NM, the planes of the triangles ADC,
ADB, make the fections BD, LN, and AC, NM, (a)
parallels. Therefore AL: LB: AN : ND (6): :
CM: MD. Which was to be demonſtrated.
PROP. XVIII. Fig. 34.
If a right-line AB be perpendicular to fome plane CD, all
the planes EF paſſing thro' that right-line AB jhall be per-
pendicular to the fame plane CD.
a 16. 11,
b 2.6.
2 31. 1.
b 8.11.
Let there be fome plane BF drawn thro' AB, making
the fection EG with the plane CD; from fome point
whereof H, (a) draw HI parallel to AB in the plane
EF; (6) then fhall HI be perpendicular to the plane CD,
and fo likewife any other lines, that are perpendicu-
lar to EG; (c) therefore the plane EF is perpendicular c 4. def. 11.
to the plane CD; and for the fame reafon any other
planes drawn thro' AB fhall be perpendicular to CD.
Which was to be demonftrated.
PROP. XIX. Fig. 35.
If two planes AB, CD, cutting one the other, are per-
pendicular to fome plane GH, their line of common fection
EF fhall be perpendicular to the fame plane (GH.)
Becauſe the planes AB, CD, are taken perpendicular
to the plane GH, it appears by 4. def. 11. that from the
point F there may be drawn in both planes AB, CD, a
perpendicular to the plane GH, which ſhall be (a) one
and the fame line, and therefore the common fection of
the faid planes. Which was to be demonflrated.
PRO P. XX. Fig. 36.
If a jolid angle ABCD be contained under three plane
angles, BAD, DAC, BAC, any two of them bowsoever
taken are greater than the third.
If the three angles are equal, the affertion is evident;
if unequal, then let the greateft be BAC; from whence (a)
a 13. 11.
a 23. 1.
take
204
The eleventh Book of
b confir.
€ 4. I.
d 20. 1.
take away BAE= BAD; and make AD AE ; and
alfo draw BEC, BD, DC.
Becauſe the fide BA is common, and AD (b) = AE ;
and the angle BAE (6)=BAD, (c) thence is BE
but BDDC is (d)
Wherefore fince AD (6)
e 5. ax. 1.
f 25. 1.
§ 4. ax. 1.
BD.
BC; therefore DC
EC.
AE, and the fide AC is com-
a 32. 1. and
fch. 31. 1.
b 20. II.
c 5. ax. I.
a 22. I.
b 23. 1.
C 4. 1.
d byp.
e 24. I.
f 20. 1.
mon, and DC - EC, (f) the angle CAD fhall be
EAC, (g) therefore the angle BAD † CAD — BAC.
Which was to be demonftrated.
PROP. XXI. Plate V. Fig. 37.
Every folid angle A is contained under lefs angles than
four plane right-angels.
For let a plane any wife cutting the fides of the
folid angle A make a many-fided figure BCDE, and
as many triangles ABC, ACD, ADE, AEB. I denote
all the angles of the polygone by X; and I term the
fum of the angels at the bafes of the triangle Y;
whereof X - 4 right-angles (@)= Y + A, but becauſe
that (of all the angles at B) (b) the angle ABE + ABC
is CBE, and the fame is true alfo of the angles
at C, at D, and at E, (c) it is manifest that Y is
X, and confequently A fhall be 4 right-angles,
Which was to be demonftrated.
PROP. XXII.
Fig. 38.
If there are three plane angles A, B, HCI, whereof tavo
howfuever taken are greater than the third, and the right-
lines which contain them are equal AD, AE, FB, &c. then
of the right-lines DE, FG, HI, connecting thofe equal right-
lines together, it is poffible to make a triangle.
A triangle may be (a) made of them, if any two
be greater than the third; but they are fo.
For (b)
make the angle HCK B, and CK CH, and draw
HK, IK; (c) thence KH= FG, and becauſe the angle
KCI(d) — A; (e) therefore KI DE, but KI (ƒ)
HỈ KH (FG ;) therefore DE HI + FG.
By the like argument any other two may be proved
greater than the third; and confequently (a) it is
poffible to make a triangle of them. Which was to
be demonftrated.
1
1
PROP
EUCLIDE's Elements.
205
PRO P. XXIII. Plate V. Fig. 39.
To make a folid angle MHIK of three plane angles
A, B, C, whereof two howsoever taken are greater than the
third. But it is necessary that those three angles be lefs
than four right-angles.
Make AD, AE, BE, BF, CF, CG, equal one
to the other; and of the fubtended-lines DE, EF,
FG (that is, of the equal lines GH, IK, KH) (a) make
the triangle HKI; about which (6) defcribe the circle
LHKI.* But becauſe AD is HL, (c) let ADq be
HLq+LMq. (d) and let LM be perpendicular to the
plane of the circle HKI; and draw HM, KM, IM.
Wherefore fince the angle HLM (e) is a right-angle. (f)
thence is MHq HLg + LMq (g) = ADq. therefore
MHAD. By the fame way of reafoning MK, MI,
AD (that is AE, AB, &c.) are equal; therefore fince
HM AD, and MI AE, and DE ()=HI, (4) the
angle A fhall be≈ HMI, (k) as likewife the angle IMK
B, (k) and the angle HMKC, wherefore a folid
angle is made at M of the three given plane angles.
Which was to be done. AD is affumed to be
But this is manifeft. For if AD be or
or HL, then
is the angle A (1) (m) or HLI. In like manner
fhall B be equal or HLK, and C or C-KLI.
wherefore A-- BC hall either equal or exceed
four right-angles, contrary to the Hyp. therefore rather
let AD be HL. Which was to be demonflrated.
PROP. XXIV. Fig. 40.
HL.
If a folid AB be contained under parallel planes, the oppofte
planes thereof (AG, BD, &c.) are like and equal parallcio-
grams.
* 21. 11.
a 22. II.
and 21. 1.
b 5.4.
* See Cla-
vius
cjcb. 47.1.
d 12. II.
e 3. def. 11.
£ 47. I.
g confir.
h conftr.
k 8. 1.
1 conftr. &
8. I.
m 21. I.
4cor.13.1.
b35. def.1.
C IO. 11.
The plane AC cutting the parallel planes AG, DB, (@) 216. 11.
makes the fections AH, DC, parallels, and for the fame
reafon AD, HC are parallels. Therefore ADCH is a
pgr. By the like argument the other planes of the pa-
rallelepipedon are (b) pgrs. wherefore fince AF is paral-
lel to HG, and AD) to HC, (c) the angle FAD fhall be
GHC, therefore becauſe AF (d)= HG, and AD (α)=
HC, and fo AF: AD: HG : HC, the triangles FAD,
GHC (g) are like and (4) equal; and confequently pgrs.
AE, HB are like and (4) equal, and the fame may be
thewn of the ruft of the oppofite planes, therefore, &c.
PROP.
r
d 34. 1.
e 7.5.
g 6.
6.6.
h
4.
I.
k 6. ax. 1.
206
The eleventh Book of
a 36. 1. and
i def. 6.
b 24. 11.
c 10. def.
II.
d 24. 11.
and 9. def.
il.
e 6. def. 5.
PROP. XXV. Plate V. Fig. 41.
If a folid Parallelepipedon ABCD be cut by a plane EF
parallel to the oppofite planes AD, BC; then as the baſe AH
is to the bafe BH, fo fhall folid AHD be to folid BHĊ.
EB1
Conceive the parallelepipedon ABCD to be extended
on either fide, and take AI AE, and BK
and put the planes IQ, KP, parallel to the planes AD;
BC; then the pers. IM, AH, and (a) DL, DG, (b) and
IQ, AD, EF, &c. are (a) like and equal (c) wherefore
the parallelepipedon AQ is AF; and for the fame
reafon the parallelepipedon BP BF; therefore the fo-
lids IF, EP are as multiple of the folids AF, EC, as
the bafes IH, KH, are of the bafes AH, BH. And if
the baſe IH be —,—, KH, (d) likewiſe fhall the
folid IE be,, EP; (e) confequently AH : BH
:: AF EC. Which was to be demonftrated.
The fame may be accommodated to all forts of prifmess
whence:
:
Coroll
à i, ii.
If any prifm whatſoever be cut by a plane parallet
to the oppofite planes, the fection fhall be a figure equal
and like to the oppofite planes.
PRO P. XXVI. Fig. 42.
Upon a right-line given AB, and at a point given in it
A, to make a folid angle AHIL equal to a folid angle gi
ven CDEF.
L
From fome point F (a) draw FG perpendicular to the
plane DCE, and draw the right-lines DF, FE, EG, GD;
CG. Make AH= CD, and the angle HAI DCE,
and AI CE; and in the plane HAI make the angle
HAK DCG, and AKCG, then erect KL per-
pendicular to the plane HAI, and let KL be GF,
and draw AL. Then AHIL fhall be a folid angle
equal to that given CDEF. For the conftruction of this
does wholly refemble the framing of that, as will ea
fily appear to any who examine it.
PROP
EUCLIDE's Elements.
207
PROP. XXVII. Plate V. Fig. 43.
Upon a right-line given AB to deſcribe a parallelepipedon
AK, like, and in like manner fituate, with a folid parallele-
pipedon given CD.
:
a 26. II.
b 12.6.
Of the plane angles, BAH, HAI, BAI, which are
equal to FCE, ECG, FCG, (a make the folid angle Aa 26.
equal to the folid angle C. Alſo (b) make FC: CE: :
BA: AH (6) and CE CG:: AH: AI (whence by
equality (c) FC: CG:: BA: AI) and finiſh the paralle-
lepipedon AK, which fhall be like to that which is
given.
c 22.5:
e 24. 11.
For by the conſtruction, the Pgr. (d) BH is like to FE, di.def.6.
and (d) HI to EG, and (d) BI to FG, and (e) fo the
oppofites of theſe to the oppofites of them: Therefore
the fix planes of the folid AK are like to the fix
planes of the folid CD; (f) and confequently AK, fg def. 146
CD, are like folids. Which was to be demonftrated.
PRO P. XXVIII. Fig. 44.
If a folid parallelepipedon AB be cut by a plane
FGCD drawn thro' the diagonal-lines DF, CG, of the
oppofite planes AE, HB, that folid AB fhall be equally
bifected by the plane FGCD.
For becaufe DC, FG, are (a) equal and parallels, (b)
the plane FGCD is a Pgr. and becaufe (a) the Pgrs. AE
HB, are equal and like, (b) alfo the triangles AFD,
HGC, CGB, DFE are equal and like. But the Pgrs.
AC, AG, are equal and like to FB and FD; there-
fore all the planes of the prifm FGCDAH are equal
and like to all the planes of the prifm FGCDEB,
and (c) conſequently this prifm is equal to that.
Which was to be demonflrated.
PROP. XXIX. Fig. 45.
Solid parallelepipedons AGHEFBCD, AGHEMLKİ,
being conflituted upon the fame bafe AGHE, and * in the
fame height, whofe infifting lines AF, AM, are placed in
the Jame right-lines AG, FL, are equal one to the
other.
a 24. 1.
b 34. 1.
cg. def. 11.
* i.e. be-
tween the
parallel
planes AG-
HE, FL
KD, & ſo
underſtand
For
it in the fols
208
The eleventh Book of
a 10.defi1
& 35. 1.
b 3. and 2.
at. 1.
a 34. 1.
b 29. 11.
CI. ex. I.
* by height
underfland
the perpen-
dicular
drawn frem
the plane of
the buje to
the oppofite
plane.
a 18. 6.
b 27.11&
12.11.
J
c 30def. 11
a
For (a) if from the equal prifmes AFMEDI, GBLH-
CK, the common prifm NBMPCI be taken away, and
the folid AGNEHP be added, the Parallelepipedon
AGHEFBCD fhall be AGHEMLKI. Which was
to be demonftrated.
PRO P. XXX. Plate V. Fig. 46.
Solid parallelepipedons ADBCHEFG, ADCBIMLK be-
ing conftituted upon the fame bafe ADBC, and in the fame
height, whofe infifting lines AH, AI are not placed in the
fame right-lines, are equal one to the other.
For produce the right-lines HEO, GFN, and LMO,
KIP; and draw AP, DO, BQ, CN; (a) then ſhall DC,
AB, HG, EF, PQ, ON be as well equal and parallel
one to the other as AD, HE, GF, BC, KL, IM, QN,
PO; (b) wherefore the parailelepipedon ADCBPONQ
fhall be equal to either parallelepipedon ADCBHEFG,
ADCBIMLK; and (c) confequently thefe two are equal
one to the other. Which was to be dem.
PRO P. XXXI. Plate VI. Fig. 1.
Solid parallelepipedons, ALEKGMBI, CPOHQDN,
being conftituted upon equal bafes ALEK, CPO, and *
in the fame height are equal, one to the other.
First, let the parallelepipedons AB, CD, have the
fides perpendicular to the bafes, and at the fide CP
being produced, (a) make the Pgr. PRTS equal and
like to the pgr. KELA. (b) and fo the parallelepipedon
PRTSQVYX equal and like to the parallelepipedon AB.
Produce O E, ND ♪, PZ, DQF, ERB, ♪ V 2, TSZ,
YXF; and draw E♣, Bɔ, ZF.
O
S.
w
The planes O ed N, CRVH, ZTYF, (c) are parallels
one to the other; (d) and the Pgrs. ALÉK, CP w O-
PRTS, PRBZ are equal. Therefore fince the paralle-
hyp. and lepipedon CD: PV (e): : Pgr. Ce (PRBZ): R∞
:: parallelepipedon PRBZQV F: PV ; the pa-
rallelepipedon CD (ƒ) fhall be PRBZQV F (g)"=
PRVQSTYX (b) --AB. Which was to bo dem.
35. 1.
e 25. 11.
£9. 5.
g 29, 11.
h
confir.
2
But if the parallepipedons AB, CD, have fides ob-
lique to the bafe, then on the fame bafes and in the
fame height place parallelepipedons whofe fides are per-
Fen-
Plate V. Facing Pag. 208.
Fig. 1.
Fig 2.
FKEK HM
Fig. 3.A Fig. 4. Fig.5.
A DFGE PN
EK HM
B
B
Fig. 6.
D'G M IL
B
K F Fig. 7.
AB
BA
H D EI
IF GI
D EI GF
ED FB B F G
CA
Fig. 8.
B CKHI MR
E A BEF D
F
-G
Q
A A
EF NHK IC
FC
Fig. 14.
I E D H
A
H F
C
Α
DFig. 9.
HB
Fig. 10.
I
Fig. 137
B
B
A Fig. 15
Fig. 16.
A
C
Fig. 11. Fig. 13.
Fig.17. Fig. 18. D
E
D
B K GE
F
D
B
Fig. 19.
€
Fig. 20
B
G
F
E
F
c
F
E
B
BHE
E
Fig 21.
D
D
B/D
H
H
A
G
B
C
BE
A
B
B
E
H
A
A
G
Fig. 23.
F G
Fig. 27.
AS
Fig. 25.
B
H
K
B
D
F ट
FG
Fig. 22.
Fig. 24.
B
D
Fig. 28.
GF
Fig. 26 D
Fig.
E
I
EV
EDF
A
Fig.31
Fig. 32.
Fig.33
F A
I
E
I
C
E
FC
A E
C
H
A
B
Fig. 29.
H
M
L
H
B
H
אב
Fig. 34.
A
E
Fig. 135.
Fig. R36.
D F
D
Fig. 37. F
K
K D
BI
M
Fig.3.8.
Fig. 39
B
H
A
I
H
F
D
D
E
B
B
H
FED
B
E
Р
C
F D
A
Fig. 40.
D
O
Fig. 416
Fig. 42
K
F
N
H M
E
G D
K
H
H
E
H
B
E A
F4
B C
Fig. 43.
F
K
K
I
C
B
I
E
[
Fig. 46.
}
H
E
D
M
F
L
M
B
N
B
F
H
A
E
N
Fig. 44.
P
Fig. 45.
}
EUCLIDE's Elements.
209
|
pendicular to the baſe. (k) They fhall be equal to one
another, and to thofe that are oblique, (m) whence alfo
the oblique parallelepipedons AB, CD are equal. Which
was to be demonftrated.
PROP. XXXII. Plate VI. Fig. 2.
Solid parallelepipedons ABCD, EFGL, of the fame height,
are one to the other, as their bafes, AB, EF.
Produce EHI, (a) and make the pgr. FIAB, and
(b) compleat the parallelepipedon FIÑM. It is clear that
the parallelepipedon FINM: (c) (ABCD) EFGL (d) : :
FI (AB): EF. Which was to be demonftrated.
PRO P. XXXIII. Fig. 3.
Like folid parallelepipedons, ABCD, EFGH, are to oné
another in triplicate ratio of their homologous fides AI, EK.
Produce the right-lines AIL, DIO, BIN, and (a) make
IL, IO, IN, equal to EK, KH, KF, (b) and ſo the pa-
rallelepipedon İXMT equal and like to the parallelepi-
pedon EFGH. (c) Let the parallepps. IXPB, DLYQ
be finiſhed. (d) 'Then ſhall be AL: iL (EK) : : DI: IO
(HK): : BỈ : IN (KF) (e); that is the pgrs. AD : DL
: DL: IX :: BỎ : IT. (ƒ) i. e. the parallepp. ABCD
DLQY :: DLQY: IXBP :: IXBP : IXMT. (g)
(EFGH) (b) therefore the proportion of ABCD to EFGH
is triplicate of the proportion of ABCD to DLQY,
(k) or of AI to EK. Which was to be demonftrated.
Coroll:
Hence it appears that if four right-lines be continu-
ally proportional, as the firft is to the fourth, fo is a
parallelepipedon defcribed on the firft to a parallelepipe-
don deſcribed on the fecond, being like and in like man-
ner defcribed.:
PRO P. XXXIV. Fig. 4.
In equal folid parallelepipedons ADCB, EHGF, the baſes
and altitudes are reciprocal (AD: EH:: EG. AC) And
folid parallelepipedons, ADCB, EHGF, whofe bafes and
altitudes are reciprocal, are equal.
First, let the fides CA, GE be perpendicular to the
bafes; then if the altitudes of the folids are equal, the
baſes alſo ſhall be equal, and the thing is clear. But if
Q
the
k
29.11.
m 1 ax. I.
a 45. i.
b 27. 11:
C 31. 11.
d 25. 11
a 3. í.
b 27. 11.
C 31. 1.
d hyp.
e 1. 6.
f 32. 11.
g confir.
h10. def...
k 1.6.
210
The eleventh Book of
a 3. I.
b
31. I..
C 32, II.
d 17. 5.
e 1. 6.
f confir.
g II. 5.
h 32. 11.
k hyp.
1 1.6.
m 32. 11.
n 9. 5.
the altitudes are unequal, from the greater EG (a) take
EI AC, and at I(6) draw the plane 1K parallel to
the baſe EH. Then
1. Hyp. AD: EH (c): : parallepp. ADCB: EHIK (♂)
:: parallepp. EHGF: EHIK (c) : : GL : IL (e) : : GÈ :
IE (ƒ) (AC); it is plain therefore (g) that AD: EH : :
GE: AC. Which was to be demonfirated.
2. Hyp. ADCB : EHIK (↳ : : AD : EH (k) : : EG :
EI (/): : GL: II. (m) :: parallepp. EHGF: EHIK; (»)
IL
wherefore the parallelepipedon ADCB EHGF.
Which was to be demonftrated.
:
Moreover, let the fides be oblique to the baſes, and
erect right parallelepipedons upon the fame bafes with the
fame altitude; the oblique parallepps. fhall be equal to
them. Wherefore fince by the firit part, the bafes and
altitudes of thoſe are reciprocal, the bafes and altitudes of
theſe alſo ſhall be reciprocal. Which was to be demon.
Coroll.
All that hath been dem. of parallelpps. in the 29, 30, 31,
32, 33, 34. Prop. does alfo agree to triangular prifms, which
are half parallelpps. as appears by Prop. 28. Therefore,
1. Triangular prifms are of equal height with their
bafes.
2. If they have the fame or equal baſes and the fame
altitude, they are equal.
3. If they are like, their proportion is triplicate of
that of their homologous fides.
4. If they are equal, their bafes and altitudes are re-
ciprocal; and if their baſes and altitudes are recipro-
cal, they are alio equal.
PRO P. XXXV. Plate VI. Fig. 5.
If there are two plane angles BAC, EDF, equal, and
from the point of those angles two righ-lines AG, DH, be
elevated on high, containing equal angles with the lines firft
given, each to his correpfondent angle (the angle GAB
HDE, and GAC HDF) and if in thofe elevated lines
AG, DH, fome points be taken, G, H; and from theſe points
perpendicular lines GI, HK, drawn to the planes BAC,
EDF, in which the angles firft given are, and right-lines AI,
DK, be drawn to the angles first given from the points 1,K,
which are made by the perpendiculars in the planes; thoſe
right-lines with the elevated lines AG, DH, fall contain
equal angles GAM, HDK.
Make
EUCLIDE's Elements.
211
a 8.11.
Make DH, AL, equal; and GI, LM parallels, and
MC to AC, MB to AB, KF to DF, KE to DE perpendi-
cular; and draw the right-lines BC, LB, LC, and EF,
HF, HE; (a) and LM perpendicular to the plane
BAC; (b) wherefore the angles LMC, LMA, LMB; b.3.def.11.
and for the fame reafon the angles HKF, HKD, HKE,
are right-angles. Therefore ALq (c) LMq + AMq
(c) = LMq + CMq - ACq (c) = LCq +ACq; (d)
therefore the angle ACL is a right-angle. Again ALq (e)
=LMq + MAq (e) = LMq + BMq+ BAq (e)
-†
BLq
- BAq; (d) therefore the angle ABL is alfo a right-
angle. By the like inference the angles DFH, DEH
are right-angles; (f) therefore AB DE, (f) and BL
EH, (f) and AC = DF, and CL = FH; (g)
wherefore alfo BC = EF; (g) and the angle ABC =
DEF, (g) and the angle ACB = DFE; (b) whence the
other right-angles CBM, BCM, are equal to the other
FEK, EFK (k) therefore CM= FK, (/) and fo alfo
AM DK; therefore if from LAq (m) = HDq be
taken away AMq DKq, (n there remains LMq
HKq; wherefore the triangles LAM, HDK are equi-
lateral one to the other; (p) therefore the angle LAM
=HDK. Which was to be demonftrated.
Coroll.
Therefore, if there be two plane angles equal, from
whoſe points equal right-lines are elevated on high con-
taining equal angles with the lines firft given, each to
each; perpendiculars drawn from the extreme points.
of thofe elevated lines to the planes of the angles firſt
given, are equal one to the other, viz. LM: HK.
PROP. XXXVI. Plate VI. Fig. 6.
If there are three right-lines DE, DG, DF proportional,
the folid parallelpp. DH. made of them, is equal to the
folid parallelpp. IN made of the middle-line DG (IL) which
is alſo equilateral, and equiangular to the ſaid parallelepi-
pedon DH.
Becauſe DE: IK (a) :: IL : DF, (b) the pgr. LK
fhall be FE, and by reafon of the equality of the
plane angles at D and I, and of the lines GD, IM,
alfo the altitudes of the parallelpps. are equal by the
preceding Coroll. (c) therefore the parallelps. are equal
one to the other. Which was to be demonftrated.
O z
PROP.
C 47. I.
d 48. I.
e 47. I.
f 26. 1.
g 4. I.
h 3. ax. 1.
k 26. 1.
1 47. 1.
m conftr.
n 47. 1.&
3. ax.
p 8. I.
a hyp.
b 14 60
C 31. II.
212
The eleventh Book of
a 33. 11.
PRO P. XXXVII. Plate VI. Fig. 7.
If there are four right-lines A, B, C, D, proportional, the
folid parallelepps. A, B, C, D being like, and in like fort de-
Jcribed from them, ſhall be proportional. And if the folid pa-
rallelpps. being like and in like fort deſcribed, be proportional
(A: B:: CD.) then those right-lines A, B, C, D, fhall be
proportional.
For the proportions of the parallelepps. (a) are tripli-
cate of thofe lines; therefore if A: B:: C: D, 6) then
b jch. 23.5. fhall the parallelpp. A: parallelpp. B:: parallelpp. C:
parallelp. D. and fo alfo contrarily.
a 12. I.
b 4. and 3.
def. 11
C 17. I.
a 34. 1.
b 29. 1.
C 4. I.
dich. 15. 1.
€ 34. I.
f 9. 11. and
I ax.
g 33. 1.
h7.1.11
PRO P. XXXVIII. Fig. 8.
If a plane AB be perpendicular to a plane AC, and a
perpendicular-line EF be drawn from a plane E in one
of the planes (AB) to the other point AC, that perpen-
dicular EF ball fall upon the common fection of the planes
AD.
If it be poffible, let F fall without the interfection
AD, and in the plane AC, (a) draw FG perpendicular
to AD, and join EG. The angle FGE (6) is a right-an-
gle, and EFG is fuppofed to be fuch alfo; therefore two
right-angels are in the triangle EFG. (c) Which is abfurd.
PROP. XXXIX. Fig. 9.
If the fides (AE, FC, AF, EC, and DH, GB, DG, HB)
of the oppofite planes AC, DB, of a folid parallelepipedon AB,
be divided into two equal parts, and planes ILQO, PKMR,
be drawn thro' their fections, the common fection of the
planes ST, and the diameter of the folid parallelepipedon
AB hall divide one the other into two equal parts.
Draw the right-lines SA, SC, TD, TB. Becauſe (a)
the fides DO, OT are equal to the fides BQ, QT, (6)
and the alternate-angles TOD, TQB equal, alfo (c) the
bafes DT, T'B, and the angels DTO, BTQ are equal,(d)
therefore DTB is a right-line, and fo in like manner is
ASC. Moreover (e) as well AD is parallel and equal to
FG (e) as FG to CB. and (f) thence AD is parallel and
equal to CB; (g) and confequently AC to DB () where-
fore AB and ST are in the fame plane ABCD. There-
fore fince the vertical-angles AVS, BVT, and the alter-
nate-
EUCLIDE's Elements.
213
VT. Which
nate-angles ASV, BTV are equal (k) and AS—BT;
therefore fhall AV be = BV, (/) and SV = VT. Which
was to be dem.
k 7. ax. 1.
125.1.
Coroll.
Hence in every parallelepipedon, all the diameters
bifect one another in one point, V.
PROP. XL. Plate VI. Fig. 10.
If two prifms ABCFED, GHMLIK, be of equal alti-
tude, whereof one hath its bafe ABCF a parallelogram, and
the other GÍM a triangle, and if the parallelogram ABCF
be double to the triangle GHM; thofe prifms ABCFED,
GHMLIK are equal.
For if the parallelepipedons AN, GQ, be compleated,
(a) they fhall be equal, becauſe of the equality (b) of the
bafes AC, GP, and (c) of the altitudes; (d) therefore alſo
the prifms, (e) the halfs thereof fhall be equal. Which
was to be dem.
b
a 31. 11.
34. 1. &
7. ax.
c hyp.
d 28. II.
I
e 7. ax. 1.
Schol.
From the preceding demonftrations, the demenfion of trian-
gular prifms, and quadrangular, or parallelepipedons, is learnt ;
viz. by multiplying the altitude into the baje.
As if the altitude be 10 foot, and the bafe 100 ſquare
foot the baſe may be meafured by fch. 35. 1. or by 41.1.)
then multiply 100 by 10, and 1000 cubic foot fhall be
produced for the folidity of the prifm given.
For as a rectangle, fo alfo is a right parallelepp. produ-
ced from the altitude multiplied into the bafe. There-
fore every parallelepipedon is produced from the altitude
multiplied into the baſe, as appears by 31. of this Book.
Moreover, fince the whole parallelepipedon is produced
from the altitude drawn into the bafe, the half thereof
(that is, a triangular prifm) fhall be produced from the
altitude drawn into half the bafe, namely the triangle.
Andr Tacq
0 3
Au
214
The eleventh Book of
An Advertiſement.
Obf. That of thoſe letters which denote a folid angle, the
firft is always at the point in which the angle is; but of thoſe
letters which denote a pyramid, the last is at the fupreme
point thereof.
Ex. gr. the folid angle ABCD is at the point A; and
the fupreme point of the pyramid BCDA is at the point
A, and the bafe is the triangle BCD.
The End of the Eleventh Book.
THE
[ 215 ]
The TWELFTH BOOK
O F
EUCLIDE's
ELEMENT S.
PROPOSITION I. Plate VI. Fig. 11, 12.
L
FM.
IKE polygonous figures ABCDE, FGHIK infcribed
in circles ABD, FGI, are one to another, as the
Squares defcribed on the diameters of the circles AL,
Draw AC, BL, FH, GM. Becauſe (a) the angle
ABCFGH, (a) and AB : BC::FG: GH(b) there-
fore fhall the angle ACB (c)(ALB) be FHG (c) (FMG.)
but the angles ABL, FGM (d) are right and fo equal;
(e) therefore the triangles ABL, FGM are equiangular. (ƒ)
wherefore AB: FG :: AL: FM. (g) therefore ABCDË :
FGHIK ALq: FMq,
:
Coroll.
:
Hence (becaufe AB: FG AL: FM :: BC: GH
c,) the ambits of like polygonous figures infcribed in a
circle are in the fame (4) proportion as the diameters.
PROP. II. Fig. 13, 14.
Circles ABT, EFN, are in proportion one to another,
as the Squares of their diameters AC, EG are.
:
Suppofe ACq: FGq: the circle ABT: I. I fay
then I is equal to the circle EFN.
For firſt, if it be poffible, let I be leſs than the circle
EFN, and let K be the excefs or difference. Infcribe
the fquare EFGH in the circle EFN, (a) it being the
half of a circumfcribed fquare, and fo greater than
the femicircle. (6) Divide equally in two the arches
a 1. def. 6.
b 6. 6.
C 21. 3.
d 31. 3.
e 32. 3.
f cor. 4. 6.
g 22.6,
h 1. 12.&
12.5:
afb. 7.4.
b 30. 3.
04
EF
216
The Twelfth Book of
cfch. 27. 3. tangent PQ (c) which
d
41. I-
e 1. 10.
f hyp. and
3. ax.
£ 30. 3. &
I.
1. paſt. I.
h 1. 12.
k hyp.
19. ax. 1.
m 14. 5.
n hyp:
0 14. 5.
Р
11. 5.
EF, FG, GH, HE, and at the points of the divifions
join the right-lines EL, LF, &c. thro' L draw the
tangent PQ (c) which is parallel to EF; and produce
HEP, GFQ, then is the triangle ELF (d) the half of the
pgr. EPQF, and fo greater than the half of the fegment
ELF; and in like mnaner the rest of thoſe triangles
exceed the halfs of the reft of the fegments. And if
the arches E L, LF, F M, &c. be again bisected,
and the right-lines joined, the triangles will likewiſe
exceed the half of the fegments. Wherefore if the
fquare EFGH be taken from the Circle EFN, and the
triangles from the other fegments, and this to be done
continually, at length (e) there will remain fomę.
magnitude leſs than K. Let us have gone ſo far, name-
ly, to the fegments EL, LF, FM, &c. taken together
lefs than K. Therefore I (ƒ) (the circle EFN-K) –
the polyg. ELFMGNHO (the circle EFN- the feg-
ment EL+LF, &c.) In the circle ABT (g) conceive
a like polygon AKBSCTDV infcribed. Therefore fince
AKBSCTDV: ELFMGNHO (b): ACq: EGq (k): :
the circle ABT: I. and the polyg. AKBSCTDV (1)
the circle ABT. the polyg. ELFMGNHO (m) fhall be
I. but before, I was ELFMGNHO, which is
repugnant.
:
Again, if it be poffible, let I be the circle EFN.
Therefore becauſe ACq: EGq (n): : the circle ABT: I;
and inverſely I the circle ABT EGq: HCq, fuppofe
I: the circle ABT: : the circle EFN: K. (0) therefore
the circle ABT — K. (p) and EGq : ACq
C
the circle
EFN: K, which was juft now fhewn to be repugnant.
Therefore it must be concluded, that I is to the
circle EFN. Which was to be demonftrated.
Coroll.
=
Hence it follows, that as a circle is to a circle, ſo
is a polygon infcribed in the first to a like polygon in-
ſcribed in the fecond.
PROP. III. Plate VI. Fig. 15.
Every Pyramid ABCD having a triangular bafe, may
be divided into two pyramides AEGH, HIKC, equal, and
like one to the other, having bafes triangular, and like to the
whole ABDC; and into two equal prifms, BFGEIH,
FGDIHK; which two priſms are greater than the half of
the whole pyramid ABDC.
Divide
EUCLIDE's Elements.
217
Alfo the trian-
Divide the fides of the pyramid into two parts at the
points E, F, G, H, I, K, and join the right-lines EF, FG,
GE, EI, IF, FK, KG, GH, HE. Becauſe the fides of
the pyramid are proportionally cut (a) thence HI, AR,
and GF, AB; and IF, DC; and HG, DC, &c. are
parallels, and confequently HI, FG; and GH, FI are
alfo parallels, therefore it is apparent that the triangles
ABI), AEG, EBF, FDG, HIK, (b) are equiangular, and
that the four laſt are (c) equal: In like manner the trian-
gles ACB, AHE, EIB, HIC, FGK are equiangular; and
the four laſt are equal one to the other.
gles BFI, FDK, IKC, EGH; and laftly, the triangles
AHG, GDK, HKC, EFI are like and equal. Moreover
the triangles, HIK to ADB, EGH to BDC, and EFI
to ADC, and FGK to ABC, (d) are parallel. From
whence it evidently follows, first, that the pyramids
ALGH, HIKC are equal, and (e) like to the whole AB-
DC, and to one another. Next, that the folids BFGEIH,
FGDIHK are prifms, and that of equal heights, as be-
ing placed between the parallel planes ABD, HIK, but
the baſe BFGE is (ƒ) double of the baſe FDG, wherefore
the faid priſms are equal; whereof the one BFGEIH is
greater than the pyramid BEFI, that is, than AEGH,
the whole than its part; and confequently the two
prifms are greater than the two pyramids and fo exceed
the half of the whole pyramid ABDC. Which was
to be dem.
PROP. IV. Fig. 16, 17.
If there are two pyramids ABCD, EFGH, of the fame
altitude, having triangular baſes ABC, 1FG; and either
of them be divided into two pyramids (AILM, MNOD;
and EPRS, STVH) equal one to the other and like to the
whole; and into two equal prifms (IBKLMN,KLCNMO;
and PFQRST, QRGTSV ;) and if in like manner either of
thefe pyrs. made by the former divifion be divided, and this
be done continually; then as the base of one pyramid is to
the base of the other pyramid, fo are all the prifms which
are in one pyramid, to all the prifms which are in the other
pyramid, being equal in multitude
For (applying the conftruction of the precedent
prop.)BC KC (a): :FG: QG; (b) therefore the triangle
ABC is to the like triangle LKC as EFG is to (c) the
like RQG; therefore by permutation ABC: EFG (d) ::
LKC
a 2.6.
b 29. 1.
c 26. I.
d 15. 11.
e 10. def.
II.
f2 ax. I.
g 40. 11.
a 15. 5.
b 22.6.
c 2.6.&c.
d16. 5.
218
The twelfth Book of
f 1. 5.
g 12. 5.
eſch. 34. 11. LKC: ROG (e): the prifm KLCNMO : QRGTSV
(for thefe are of equal altitude) (f): IBKLMN: PF-
QRST; (g) wherefore the triangle ABC: FFG: the
prifm KLCMNO IBKLMN the prifm QRGTSV÷
PFQRST. Which was to be dcm.
a 1. 10.
b 4. 12.
£ hyp.
d 14. 5.
:
But if the pyramids MNOD, AILM; and FPKS,
STVH be further divided; in like manner the four
new prifmns made hereby fhall be to four produced be-
fore, as the bafes MNO and AIL are to the bafes STV,
and EPR, that is, as LKC to RQG, or as ABC to EFG.
(b) wherefore all the prifms of the pyramid ABCD
are to all the prifms of the pyramid EFGH as the
bafe ABC is to the bafe EFG. Which was to be dem.
PROP. V. Plate VI. Fig. 18, 19.
Pyramids ABCD, EFGH, being of the fame altitude,
and having triangular baſes ABC, EFG, are one to another
as their bafes ABC, EFG.
Let the triangle ABC: EFG :: ABCD: X. I fay
X is equal to the pyramid EFGH. For if it be poffible,
let X be EFGH. and let the excefs be Y. Divide
the pyramid EFGH into prifms and pyramids, and
the other pyramids in like manner, (a) till the pyrs.
left EPRS, STVH, be less than the folid Y. Therefore
fince the pyramid EFGH = X + Y, it is manifeſt
that the remaining prifms PFQRST, QRGTSV are
greater than the folid X. Conceive the pyramid ABCD
divided after the fame manner; (b) then will be the
prifm IBKLMN KLCNMO: PFQRST+QRGTSV
:: ABC: EFG (c) : : the pyr. ABCD: X; (d) there-
fore X the prifm PFQRST + QRGTSV; which is
contrary to that which was affirmed before.
→
Again, conceive X the pyr. EFGH, and make the
pyr. EFGH: Y:X: the pyr. ABCD (e) : : EFG :
e hyp.and cor. ABC Becauſe EFGH (f) X, (g) thence Y
pyr. ABCD; which is fhewn
Therefore I conclude, that X is
Which was to be demonftrated.
4.5.
I suppos
$ 14. 5,
the
before to be impoſſible.
equal to the pyr, EFGH.
PROP.
EUCLIDE'S Elements.
219
PRO P. VI. Plate VI. Fig. 20, 21,
Pyramids ABCDEF, GHIKLM, being of the fame al-
titude, and having polygonous bafes ABCDE, GHIKL,
are to one another as their bafes ABCDE, GHIKL. are.
Draw the right-lines AC, AD, GI, GK, then is the
baſe ABC: ACD) (a): : the pyr. ABCF: ACDF; (b)
therefore by compofition, ABCD: ACD: the pyr.
ABCDF: ACDF; (a) but alſo ACD: ADE: : the
pyr. ACDF: ADEF; (c) therefore by equality ABCD
·ADE: : ABCDF: ADÉF, and (b) thence by compo-
fition ABCDE: ADE: : the pyr. ABCDEF: ADEF;
moreover ADE: GKL (d): the pyr. ADEF: GKLM;
and as before, and inverfely GKL GHIKL:: the pyr.
GKLM: GHIKLM; (c) therefore again by equality
ABCDE: GHIKL:: the pyr. ABCDEFGHIKLM.
Which was to be demonftrated.
If the bafes have not fides of equal multitude, the
demonſtration will proceed thus. The bafe (Fig. 20, 22.)
ABC: GHI (e): : the pyr. ABCF : GHIK (e) and ACD
GHI: the pyr. ACDF: GHIK, (f) therefore the baſe
ABCD: GHI :: the pyr. ABCDF: GHIK. (e) More-
over the baſe ADE: GHI :: the pyr. ADEF: GHIK,
(f) therefore the baſe ABCDE: GHI; the pyr.
ABCDEF : GHIK.
PROP. VII. Fig. 23.
Every prifm, ABCDEF, having a triangular bafe, may
be divided into three pyrs. ACDF, ACBF, CDFE, equal
one to the other, and having triangular baſes.
Draw the diameters of the parallelograms AC, CF,
FD. Then the triangle ACB is (a)=ACD ; (b) therefore
the pyramids of equal height ACBF, ACDF, are equal.
In like manner the pyr. DFAC➡the pyr. DFEC, but
ACDF and DFAC are one and the fame pyr. (c) there-
fore the three pyramids ACBF, ACDF, DFEC, into
which the priſm is divided, are equal one to the other.
Which was to be demonftrated.
Hence, every pyramid is the third part of the prifm
that has the fame bafe and height with it, or every
prifm is treble of the pyramid that has the fame bafe and
height with it.
For
a 5..12.
b 18 5.
C 22. 5.
d 5. 12.
e 5.12.
£ 24. 5:
a 34. i.
b
5.12.
C 1. ax. Iv
220
The twelfth Book of
27. 12.
b 1.5.
a 27. 11.
b
C 28. II.
For refolve the polygonous prifm ABCDEGHIKF,
(Fig. 24.) into triangular prifms; and the pyr. ABCD-
EH into triangular pyramids; (a) then all the parts of
the prifm fhall be treble to all the parts of the pyramids,
(b) confequently the whole prifm` ABCDEGHIKF is
treble to the whole pyr. ABCUEH. Which was to be
demonjirated.
PRO P. VIII. Plate VI. Fig. 25, 26.
Like pyramids ABCD, EFGH, which have triangular
bales ABC, EFG are in triplicate ratio of their homolo-
gous fides AC, EG.
(a) Compleat the parallelpps. ABICDMKL, EFNG-
9. def. 11. HQOP, which (b) are like, and (c) fextuple of the pyra-
mids ABCD, EFGH; (d) and therefore the pyrs. have
the fame proportion to one another as the parallelps.
have, that is, (e) triplicate of their homologous fides.
and 7. 12.
d 15. 5.
e 33. 11.
a 28. 11.
and 7. 12.
b 34. 11.
C 15.5.
d hyp.
e 15. 5.
£ 34. 11.
£ 6.
60.
ax. I.
Coroll.
Hence, alfo like polygonous pyramids are in tripli-
cate ratio of their homologous fides; as may be eaſily
prov'd by refolving them into triangular pyramids.
PROP. IX. Fig. 25, 26.
In equal pyramids ABCD, EFGH, having triangular
baſes ABC, EFG, the bafes and altitudes are reciprocal ;
And pyramids having triangular bafes, whofe altitudes and
bafes are reciprocal, are equal.
1. Hyp. The compleated parallelpps. ABICDMKL,
EFNGHQOP are (a) fextuple of the equal pyramids
ABCD, EFGH (each to each) and fo equal one to the
other; therefore as the altitude (H): the altitude (D)
(6) :: ABIC: EFNG; (‹) ; : ABC : EFG. Which
was to be demonftrated.
2. Hyp. The altitude (H): the altitude (D.) (d) : :
ABC: EFG (e) :: ABIC: EFNG (f) therefore the
parallel pps, ABICDMKL, EFNGHOOP are equal;
(g) confequently alfo the pyramids ABCD, EFGH being
fubfextuple of the fame, are equal. Which was to be
demonftrated.
The fame is applicable to polygonous pyramids, for they
may alſo in like manner be reduced to triangular.
Coroll
EUCLID E's Elements.
221
Coroll.
Whatſoever is dem. of pyramids in prop. 6, 8, 9 does
like-wife agree to any fort of prisms; feeing they are triple
of the pyramids that have the fame bafe and altitude with
them. Therefore
1. The proportion of prifms of equal altitude is the
fame with that of their baſes.
2. The proportion of like prifms is triplicate of
that of their homolgous fides.
3. Equal prifins have their bafes and altitudes reci-
procal; and prifms which are ſo reciprocal, are equal.
Schol.
From what has been hitherto dem. the dimenfion of
any prifms and pyramids may be collected.
(a) The folidity of a prifm is produced from the al-
titude multiplied into the baſe; () and therefore like-
wife that of a pyr. from the third part of the altitude
multiplied into the bafe.
PROP. X, Plate VI. Fig. 27.
Every Cone is the third part of a cylinder having the fame
bafe with it ABCD, and the altitude equal.
a'i.cor.12.
& jcb. 4•
II.
b 7. 12.
a ſch. 7. 4•
and cor. 9.
12.
and cor. 9.
12.
If you deny it, then firſt let ſuch cylinder be more
than triple to the cone, and let the excefs be E. A priſm
defcribed on a fquare in the circle ABCD (a) (Fig. 13,
14.) is fubduple of a prifm deſcribed upon a fquare about
the circle, being equal to it and the cylinder in height.
Therefore a prifm upon the fquare, ABCD exceeds the
half of the cylinder, and likewife a prifin upon the baſe
AFB, of equal height to the cylinders, (b) is greater than bſcḥ.27.3.
the half of the fegment of the cyl. AFB, continue an
equal bifection of the arches, and ſubſtract the prifms till
the remaining fegments of the cyl. namely, at AF, FB,
&c. become less than the folid E. Therefore the cyl.
- fegm. AF, FB, &c. (the priſm on the baſe AFBGCH-
DI) (c) is greater than the cyl. E (d) (the triple of the
cone) therefore the pyr. (e) a third part of the taid prifm
(being placed on the fame bafe, and of the fame height)
is greater than the cone of equal height on the baſe
ABCD a circle, i. e. the part greater than the whole
Which is abfurd.
But
c 5. ax. 1.
d hyp.
e cor.7.12.
222
The twelfth Book of
But if the cone be affirmed to be greater than the
third part of the cyl. then let the excefs be E. Subtract
the pyrs. from the cone, as you did in the first part
the prifms from the cyl. till fome fegments of the cone
remain, fuppofe AF, FB, BG, &c. lefs than the folid
E, therefore the cone - E (f) of the cyl.) the pyr.
AFBGCHDI (the cone feg. AF, FB, &c.) therefore
the prifm triple to the pyr. (viz. of equal height, and
on the fame baſe) is greater than the cyl. on the baſe
ABCD), the part than the whole. Which is abfurd.
Wherefore it must be granted, that the cyl is equal to
triple of the cone. Which was to be dem.
PROP. XI. Plate VI. Fig. 28. 29.
Cylinders and Cones ABCDK, EFGHM, being of the
fame altitude, are to one another as their bafes ABCD,
EFGH are.
N
Let the circle ABCD : the circle EFGH: the cone
ABCDK: N. I fay N is equal to the cone FFGHM.
For if it be poffible, let Ñ be the cone EFGHM,
and let the excefs be O. The preparation and argumen-
tation of the prec. prop. being fuppofed; then fhall O
be greater than the fegments of the cone EP, PF, FQ,
&c. and fo the folid Ń the
the pyr. EPFQGRHSM. In
and the circle ABCD (a) make a like polyg. fig. ATBVCX-
DY. Becauſe the pyr. ABVYK: the pyr. EFQSM (b) : :
the polygon ATBVY: the polygon EPFQS (c): : the
circle ABCD : the circle EFGH (d): the cone ABCDK
: N (e) thence the pyr. EPFQGRHSM fhall be — N,
contrary to what was affirmed before. Again, conceive N
a 30. 3.
I post.
b 6.12.
c cor, 2. 12.
d hyp.
e 14. 5.
f hyp. and by
inverfion.
g 14. 5.
the cone EFGHM, and make the cone FFGHM :
O:: N: the cone ABDCK (ƒ): the circle EFGH:
ABCD; (g) therefore O the cone ABCDK; which
is abfura, as appears by what is fhewn in the first part.
Therefore rather admit ABCD: EFGH :: the cone
ABCDK : EFGHM. Which was to be dem.
The fame may be dem. of cylinders, if cylinders and
prifms be conceived in the place of cones and pyramids,
therefore, &c.
Schol.
Hence, is gathered the dimenfion of all forts of cylinders
and cones. The folidity of a right cyl. is produced from
the circular baſe (a) (the dimenfion whereof is to be
demenfion.cir, learnt out of Archimedes) multiplied into the height; (b)
whence in like manner that of every cylinder.
a 1. prop de
b 11. 12.
(c) Therefore
EUCLIDE's Elements.
223
(c) Therefore the folidity of a cone is produced from e 1o. 12.
the third part of the altitude multiplied into the baſe.
PROP. XII. Plate VI. Fig. 30, 31.
Like comes and cylinders ABCDK, EFGHM, are in tri-
plicate ratio of that of the diameters TX, PR, of their baſes
ABCD, EFGH.
PR. I fay N is
fible let N be
therefore N
:
Let the cone A have to N a triplicate ratio of TX to
the cone EFGHM. For if it be pof-
EFGHM, and let the excefs be O,
the pyr. EPFQGRHSM. Let the axes
of the cones be IK, LM, and join the right-lines VK,
CK, VI, CI, and QM, GM, QL, GL. Becauſe the cones
are like, (a) thence VI: IK :: QL: LM; but the an-
gles VIK, QLM (b) are right-angles, (c) therefore the
triangles VIK, QLM are equiangular, (d) whence VC :
VI:: QG: QL. alfo VI: VK:: QL QM. therefore
by equality VC: VK:: QG: QM. (e) moreover VK:
CK:: QM: MG; therefore again by equality VC:
CK :: CG: GM; (f) therefore the triangles VKC,
QMG are like; and by a like way of reafoning the
other triangles of this pyr. are like to the other of
that, (g) wherefore the pyrs. themſelves are like. (b)
But theſe are in triplicate proportion of that of VC to
QG, (k) that is, of VI to QL, (/) or TX to PR; (m)
therefore the pyr. ATBVCXDYK: the pyr. EPFQG-
RHSM the cone ABCDK :N; (7) whence the pyr.
EPFQGRHSM N. which is repugnant to what
was affirmed before.
Again, take N the cone EFGHM, make the cone
EFGHM: 0:: N: the cone ABCDK (0): : the pyr,
EPRM: ATCK (P): GQ: VC thrice :: (2) PR: TX
thrice, therefore O (r) is
(r) is ABCDK. which was
before fhewn to be repugnant. Wherefoe N the
cone EFGHM. Which was to be demonftrated.
But forafmuch as what proportion foever cones have,
alfo cylinders, being triple of them, have the fame;
therefore cyl. fhall be to cyl. in triplicate ratio of the
diameters of their bafes.
a24. def.11
b18.def.11.
c 6.6.
d 4. 6.
e 7. 5.
f 5. 6.
g9. def. 11.
h cor.8.12.
k 4. 6.
1 15.5.
m hyp. and
11. 5.
n 14. 5.
● before &
inverfly.
p cor. 8.12
9 4.6.
I 14. 5.
PROP.
224
The twelfth Book of
2 3. I.
b 11. 12.
C II. 12.
d 6. def. 5.
a 11. 12.
b 13. 12.
* apply 9. &
7. 12.
a 14. 12.
b conſtr.
c hyp.
dii. 12.
PROP. XIII. Plate VI. Fig 22.
If a cyl. ABCD be divided by a plane EF parallel to the
oppofite planes BC, AD, then as one of cyl. AEFD is to the
other cyl. EBCF, fo is the axis GI to the axis IH.
The axis being produced, (a) take GK GI, and
HL= IH LM, and conceive planes drawn at the
points K, L, M, parallel to the circles AD, BC, (b) there-
fore the cyl. ED the cyl. AN, and the cyl. EC (b) =
BO (b)
OP; therefore the cyl. EN is the fame multi-
ple of the cyl. ED as the axis IK is of the axis IG, and
in like manner the cyl. EP is the fame multiple of the
cyl. BF, as the axis IM is of the axis IH; but as IK is
C, JIM, (c) fo is the cyl. EN —,—, — EP
(d) therefore the cyl. AEFD: the cyl. EBCF.: GI :IH,
Which was to be dem.
PRO P. XIV. Fig. 33.
Cones AEB, CFD, and cylinders AH, CK, ftanding
upon equal bafes AB, CD, are to one another, as their
altitudes ME, NF
The cyl. HA, and the axis EM, being produced,
take MLFN; and thro' the point L draw a plane
parallel to the baſe AB, (a) then fhall the cyl. AP
be=
CK, (b) but the cyl. AH: AP (CK):: MË: ML
(NF.) Which was to be dem.
The fame may be affirmed of cones which are fubtri-
ple of cylinders; as alfo of prifms and pyramids.
*
PROP. XV. Fig. 34.
In equal cones BAC, EDF and cylinders BH, EK,
the bafes and altitudes are reciproc. (BC: EF :: MD: LA)
And cones and cylinders, whofe bafes and altitudes are re-
ciprocal, are equal one to the other.
If the altitudes be equal then the bafes are equal too,
and the thing is evident. If unequal, then take away
MOLA.
1. Hyp. Then is MD: MO(a) (LA) (b): the cyl. EK
: (c) (BH) EQ_(d) : : the cir, BC : EF. Which was to
be dem.
2. Hyp.
EUCLIDE's Elements.
225
:
2. Hyp. BC: EF (e) :: DM: OM (LA) (f):´: the
cyl. EK: EQ (g): BC: EF (b): BH: EQ (k) There-
fore the cyl. EK: BH. Which was to be dem.
The fame argument may be uſed for cones.
PROP. XVI. Platé VI. Fig. 35.
Two unequal circles ABCG, DEF, having the fame cen-
ter M, to inſcribe in the greater circle ABCG a polygonous
figure of equal and even fides, which shall not touch the
Leffer circle DEF.
Thro' the center M draw the line AC cutting the cir.
DEF in F, from whence raife a perpendiculer FH, (a)
divide the femicircle ABC into two equal parts; and
the half thereof BC alfo ; and fo do continually (b) till
the arch IC becomes less than the arch HC; from I let
fall the perpendicular IL. It is manifeft that the arch
IC meaſures the whole circle, and that the number of
arches is even, and fo that the fubtended-line IC is the
fide (c) of the polygon that may be infcribed without
touching the leffer circle DEF. For HG (d) touches
the circle DEF, (e) to which IK is parallel, and placed
outwardly; (f) wherefore IK does not touch the cir
cle DEF; much lefs do CI, CK, and the other fides
of the polygon more remote from the center. Which
was to be done.
Coroll.
Obferve that IK touches not the circle DEF.
PROP. XVII. Fig. 36.
Two Spheres ABCV, EFGH, confifting about the fame
center D, being given, to infcribe a folid of many fides (or
Polyedron) in the greater Sphere ABCV, which shall not
touch the fuperficies of the leffer fphere EFGH:
Let both the fpheres be cut by a plane paffing thro'
the center, making the circles EFGH, ABCV; and the
diameters AC, BV drawn, cutting perpendicularly. In
the circle ABCV, (a) infcribe the equilateral poly-
gon VMLNC, &c. not touching the circle EFGH: Then
draw the diameters Na, and erect DO perpendicular
to the plané ABC thro' DO; and thro' the diameters
AC, Na conceive planes DOC, DON erected, which
fhall be (b) perpendicular to the circle ABCV, and fo in
P
the
é hyp.
f 14. 12.
gi..
hit. 12
k 9. 5.
a 30. 3.
b i. to.
cſch: 16. 4.
d cor.16.3.
e 28. I.
f
£ 34. def.1.
à 16.12.
b 18: 11.
226
The Twelfth Book of
d 4. 1.
cor. 33. 6. the fuperficies of the fphere make (c) the quadrants DOC,
DON. In which let the right lines CP, PQ, QR, RO,
NS, ST, T2, 70 (d) be fitted, equal, and of equal mul-
titude with CN, NL, &c. make the fame conſtruction in
the other quadrants OL, OM, &c. and in the whole
ſphere. Then I fay the thing required is done.
e 38. 11.
£ 12.ax.
g 27. 3.
n 32. 1.
k confir.
1 26. I.
m 3. ax. 1.
17.5,
0 2.6.
02.
p
6. II.
9 33. 1.
r 9. 11.
f 7. 11.
t 2. II.
u II. II.
x 4.6.
y 14. 5.
z 3 def. 11.
a 15. def. 1.
b 47.1.
c15, def. 1.
d confir.
e 28.3.
33. 6.
f
14 69
g 12. 2
h
k
32. I.
9. ax. 1.
1 5. 1.
n 19..I.
0 47. I.
P 47. I.
q cor. 16,
$2.
I 47. I,
From the points P, S, to the plane ABCV draw the
perpendiculars PX, SY, (c) which fhall fall on the fections
AC, Na. Therefore becaufe both (f the right-angles
PXC, SYN, (g) and PCX, SNY infifting on) equal
circumferences, (f) are equal, the triangles alto PCX,
SNY (b) are equiangular. Wherefore fince PC(k)—SN,
(1) alfo is PX SY, (/) and XC YN; (m) whence DX
DY, (2) and therefore DX: XC :: DY. YN; (0)
therefore YX, NC are parallels, but becauſe PX, SY are.
equal, and fince being perpendicular to the fame plane.
ABCV, they are alfo (p) parallels, (q) therefore YX, SP.
fhall be equal and parallel, () whence SP, NC, are pa-
rallel one to the other; and fo the () quadrilateral NC-
PS, and for the fame reafon SPQT, TQRG, as alfo the
(t) triangley RO are fo many planes. In like manner.
the whole ſphere may be fhewn full of fuch quadrilaterals.
and triangles, wherefore the figure infcribed is a po-
lyedron.
;
From the center D (u) draw DZ perpendicular to the
plane NCPS; and join ZN, ZC, ZS, ZP. Becauſe DN
NC (x): DY: YX, thence NC is (y)YX (SP.) and
in like manner SPTQ, and TQ2 R. And becauſe
the angles DZC, DZN, DZS, DZP (≈) are right, and
the fides DC, DN, DS, DP, (a) equal, and DZ common,
(b) thence ZC, ZN, ZS, ZP are equal one to the other
and confequently about the quadrilateral NCPS, (c) a cir-
cle may be defcribed, in which (becauſe NS, NC, CP,
are (d) equal, and NCSP) NC (e) fubtends more than
a quadrant, (f) therefore the angle NZC at the center is
obtufe, (g) therefore NCq2 ZCq (ZCq +ZNq).
Let NI be drawn perpendicular to AC, therefore fince
the angle ADN (b) DNC -|- DCN (k) is obtuſe, the half
of it DCN fhall be greater than the half of a right an-
gle; and fo that which remains of the right-angle CNI
fhall be lefs than it, (n) whence INIC, therefore
NCq (NIq - ICq) (0) 2 INq. therefore IN — ZC,
and confequentlyDZ(p) DI; but the point I is (9) with-
out the ſphere EFGH, and ſo, much more, the point Z..
wherefore the plane NCPS, (of which () the neareſt point
to the center is Z.) does not touch the fphere, EFGH.
And
EUCLIDE'S Elements.
227
And if a perpendicular D bo drawn to the plane SPOT,
the point, and fo alfo the plane SPQT is yet further
removed from the center, which is alſo true of the other
planes of the polyedron. Thererefore the polyedron
ORQPCN, &c. infcribed in the greater sphere, does
not touch the leffer. Which was to be done.
Coriall
Hence it follows, that if in any other sphere a folid polye-
dron, like to the aboveſaid ſolid polyedron, be infcribed, the
proportion of the polycdron in one ſphere to the polyedron in the
other is triplicate of that of the diameters of the Spheres.
For if right-lines be drawn from the centers of the
fpheres to all the angles of the bafes of the faid polye-
drons, then the polyedrons will be divided into pуrs.
equal in number and like; whofe homo. fides are femi-
diameters of the fpheres; as appears, if the leffer of
theſe ſpheres be conceived defcribed within the greater
about the fame center. For the right-lines drawn from
the center of the ſphere to the angles of the bafes will
agree one to the other by eaſon of the likeneſs of the
bafes; and fo will like pyramids be made. Wherefore
fince every pyr. in one ſphere to every pyr. like it in the
other fphere (a) has proportion triplicate to that of the
homologous fides, that is, of the femidiameter of the
fpheres; and (b) as one pyr. is to one pyr. fo all the
pyrs. that is, the folid polyedron compofed of theſe, are
to all the pyrs. that is, the folid polyedron compofed of
the others; therefore the polyedron of one fphere fhall
have to the polyedron of the other fphere, proportion
triplicate to that of the femidiameters, (c) and fo of the
diameters of the ſpheres.
PRO P. XVIII. Plate VI. Fig. 38.
Spheres BAC, EDF, are in triplicate ratio of their dia-
meters BC, EF.
}
acor.8.12.
b 12.5.
C
c 15. 5.
Let the fphere BAC be to the ſphere G in tripli. pro-
portion of that of the diameter BC to the diameter EF.
I ſay G➡EDF, For if it be poffible, let G be → EDF,
and conceive the fphere G concentrical with EDF. In
the ſphere EDF (a) infcribe a polyedron not touching
the fphere G, and a like polyedron in the fphere BAC.
Thefe poleydrons (6) are in triplicate proportion of the bor.17.12.
P 2
diameters
a 17. 12.
228
The twelfth Book of
c byp:
d 14, 5.
diameters BC, EF, (c) that is, of the ſphere BAC to G.
(d) confequently the fphere G is greater than the polye-
dron infcribed in the ſphere EDF, the part than the
whole.
Again, if it be poffible, let the ſphere G be EDF.
and as the ſphere EDF is to another fphere H, fo let G
e hyp inverfe. be to BAC, (e) that is, in triplicate proportion of the dia-
f 14. 5.
meter EF to BC, therefore fince BAC (f)H, we ſhall
incur the abſurdity of the first part, wherefore rather the
ſphere G=EDF. Which was to be demonftrated.
Coroll.
Hence, as one fphere is to another ſphere, fo is a
polyedron defcribed in that to a like polyedron de-
fcribed in this..
The End of the Twelfth Book.
1
THE
[229]
The THIRTEENTH BOOK
O F
EUCLIDE's
ELEMEN T S.
I'
PROPOSITION I. Plate VI. Fig. 39.
Fa right-line z be divided according to extreme and
mean proportion (z: a:: a: e.) the square of the half
of the whole line z, and of the greater fegment a, as
one line, is quintuple to that which is defcribed of half of
that whole line z.
I fay Q. a + z = 5 Q: 1 z.
1 5 Q: z. (a) that is aazz
za =zz + 2 zz (b) or aa + zazz. For zeza (c)
zz. and ze (d) aa. (e) therefore aa za
Which was to be demonftrated.
PROP. II. Fig. 39.
+
zz.
If a right-line za be in power quintuple to a fegment
of it felfz. the line double of the faid fegment (z) being
divided according to extreme and mean proportion, the
greater fegment is (a) the other part of the right-line at
first given z -- a.
fay z: aa: e. Becauſe by the hyp. * aa - zz
+zazz zz; or aa+zazz (a)
=ze +
za, (b) thence fhall aa = ze. (c) wherefore z; a :
Which was to be demonftrated.
a 4. 2.
b
3. ax. I.
C 2. 2.
d hyp, and
16.6.
e 2. ax. &
1.ax.
*
4. 2.
:: a : e.
a 2.2.
b
<
§‹ ax.
C
P 3
PROP.
230
The thirteenth Book of
1
a 4. 2.
b 3. ax.
C 3.2.
d byp. and
17.6.
a 4. 1.
b 3. 2.
€ 17.6.
d 2. ax:
a byp.
PROP. III. Plate VI. Fig 40.
*
If a right-line z be divided according to extreme and
mean proportion (z: a::a: e.) the line made of the less
fegment e and half of the greater fegment a, is in pover
quintuple to the fquare, which is defcribed of the half line
of the greatest fegment a.
I fay Q: e+
et a
aaca
+
f
For ee
5 Q¹a: (a) that is ee
aaaa. (b) or eeeaaa.
-|- ea (c) === ze (d) = aa. Which was to be demonftrated.
PROP. IV. Fig. 41.
1
If a right-line z be cut according to extreme and mean
proportion (za: : a:e.) the ſquare made of the whole line z,
and that made of the leffer fegment e, both together, are
triple of the fquare made of the greater fegment a.
I fay zzee 3 aa. (a) or aa --ee - 2 ae + ee
3 aa. For aẹ + ee (6)= ze (e) aa. (d) therefore aa
+ 2 ae | zee=
3 aa.
Which was to be demonfirated.
D
A
A
PROP. V.
CB If a right-
If a right-line AB be cut
according to
extreme and
mean proportion in C, and a line 'AD, equal to the greater
Segment BC, added to it, the whole right-line DB is di-
vided according to extreme and mean proportion; and the
greater fegment is the right-line AB given at the begin-
ning.
For becauſe AB: AD. (a) :: AC: CB. and by inverfion
AD:AB:: CB: AC. therefore by compofition DB: AE
: : AB : AC (AD.) Which was to be demonftrated.
But if BD: BA ::
::AD: BA - AD.
Schol.
BA: AD. then fhall be BA: AD.
For by dividing BD-BA (AD):
BA: :
BA — AD : AD. therefore inverſely BA : AD: :
AD: BA - AD.
I
PROP.
EUCLIDE's Elements.
231
PROP. VI,
D
A.
C -B If a rational right-line
AB be cut according to ex-
treame and mean proportion in C, either of the jegments (AC,
CB) is an irrational-line of that kind which is called
apotome or refidual.
5
AB. (b)
DAq.
To the greater fegment AC (a) add AD
therefore DCq= DAq. (c) therefore DCq
confequently (d) fince AB, (e) and fo the half thereof
DA are p, likewife DC is p. But becaufe 5:1: not
Q:Q(f) thence is DC DA. g) therefore DC-AD,
that is, AC, is a refidual-line. Further, becauſe ACq
(b)=ABxBC, and AB is p, (i) likewiſe BC is a refidual-
line. Which was to be demonstrated.
PROP. VII.
Plate VI. Fig. 42.
If three angles of an equilateral pentagon ABCDE, whe-
ther they follow in order, (EAB, ABC, BCD,) or not,
(EAB, BCD, CDE) are equal, the pentagon ABCDE fhall
be equiangular.
Let the right-lines BE, AC, BD, be fubtended to the
equal angles in order.
Becauſe the fides EA, AB, BC, CD, and the included
angles (a) are equal, (b) therefore ſhall the baſes BE, AC,
BD, (c) and the angles AEB, ABE, BAC, BCA, be equal,
(d) Wherefore BFFA, (e) and confequently FC=
FE; therefore the triangles FCD, FED, are equilateral
one to the other: (f) whence the angle FCD-FED. (g)
confequently the angle AEDBCD. In like manner
the angle CDE is equal to the reft; wherefore the pen-
tagon is equiangular. Which was to be demonftrated.
But if the angles EAB, BCD, CDE, which are not
in order, be fuppofed equal, (b) then ſhall the angle AEB
BDC, and BE =BD. (k) and thence the angle BED
=BDE. (1) confequently the whole angle AED CDE,
therefore becauſe the angles A, E, D, in order, are equal,
as before, the pentagon fhall be equiangular. Which
was to be demonftrated.
P 4
PROP.
a 3. I.
b 1. 13.
c 6. 10.
d byp.
e ſch. 12.
10.
f 9. 10.
8 74. 10.
h 17. 6.
i 98. 10.
a hyp.
b 4. 1.
c 4. and 5.
I.
d 6.1.
e 3. ax. 1.
f 8. 1.
g 2. ax. I.
h 4. 1.
k 5. 1.
12.4x-
232
The thirteenth Book of
2 14-4.
b 28.3.
c 27.3.
d 32. 1.
e 33. 6.
f 6. 1.
g 27. 3.
h 4. 6.
a lop, and
27.3.
b
32. I.
C 7. ax. 1.
d 5. I
•
€ 1. ax. I.
£4. 6.
8 cor. 15.4.
PRO P. VIII. Plate VI. Fig. 43.
If in an equilateral and equiangular pentagon ABCDE,
tavo right-lines BD, CE, fubtend two angles BCD, CDE
following in order, thofe lines cut one another according_to
extreme and mean proportion; and their greater fegments BF
or EF are equal to the fide of the pentagon BC.
(a) Defcribe about the pentagon the circle ABD. (b)
The arch ED is BC, (c) therefore the angle FCD
FDC. (d) therefore the angle BFC = 2 FCD (FCD +
FDC.) But the arch BAE is = 2 ED, and confequently
the angle BCF (e) 2 FCD=
— 2 FCD — BFC. (f) wherefore
=BC. Which was to be demonftrated. Moreover,
becauſe the triangles BCD, FCD, are (g) equiangular. (b)
therefore BD DC (BF); : CD: (BF) FD. and likewiſe
EC: EF:: EF: FC. Which was to be demonftrated.
PROP. IX. Fig. 44.
If the fide of an Hexagon BE, and the fide of a Decagon
AB both defcribed in the fame circle ABC, be added together,
the whole right-line AE is cut according to extreme and
mean proportion (AE: BE:: BE: AB.) and the greater
Jegment therefore is the fide of the Hexagon BE.
Draw the diameter ADC, and join the right-lines
DB, DE. Becauſe the angle BDC (a)=4 BDA and the
angle BDC (b)= 2 DBĂ (DAB+DBA) thence fhall
IBA (6)(DBÉ - BED) (c) be=2 BDA (d) 2 BDE,
whence the angle DBA or DAB (e)= ADE. There-
fore the triangles ADE, ADB, are equiangular : (f)
wherefore AE AD (g) (BE) :: AD: (BE) AB.
Which was to be demonftrated.
:
Coroll.
Hence, if the fide of a hexagon in a circle be cut
according to extreme and mean proportion; the greater
fegment thereof shall be the fide of the decagon in the
fame circle.
PROP
EUCLIDE's Elements.
233
PROP. X. Plate VII. Fig. 1,
If an equilateral Pentagon ABCDE be infcribed in a
tircle ABCE, the fide of the pentagon AB containeth in
porver both the fide of a hexagon FB, and the fide of a
decagon AH infcribed in the fame circle.
Draw the diameter AG, and bifect the arch AH in
K, and draw FK, FH, FB, BH, HM.
The femicircle AG the arch AC (a) AG—AD.
that is, the arch CG=GD (b) = AHHB. therefore
the arch BCG = 2 BHK ; (c) and fo the angle BFG
2 BFK. (d) but the angle BFG = 2 BAG; (e) therefore
the angle BFK = BAG. Wherefore the triangles BFM,
FAB, (f) are equilangular; (g) whence AB: BF ::
BF: BM; (b) therefore AB X BMBFq. Moreover
the angle AFK (k)= HFK, and FA=FH; (m) where-
fore AL=LH, (m) and the angles FLA, FLH are equal,
and fo right-angles, therefore the angle LHM (m) =
LAM (2) = HBA; therefore the triangles AHB, AMH,
(0) are equiangular; wherefore AB: AH: AH: AM;
(9) therefore AB × AM AHq. Since therefore ABq
() ABX BM + AB X AM, () thence ABq BFq
+AHq. Which was to be demonftrated.
Coroll.
=
1. Hence, a right-line (FK) which being drawn from
the center (F) divides an arch (HA) into two equal feg-
ments, does alfo divide the right-line (HA) fubtending
that arch perpendicularly into two equal fegments.
2. The diameter of a circle (AG.) drawn from any
angle (A) of a pentagon, does divide equally in two,
both the arch (CD,) which the fide of the pentagon
oppofite to that angle fubtends, and alfo the oppofite
fide it felf (CD) and that perpendicularly.
Schol. Fig. 2.
Here, according to our promife, we shall lay down a
ready praxis of the 11th prop. of the 4th Book.
Problem
To find the fide of a pentagon to be infribed in a circle
ADB.
a 28. 3. &
3 ax.
b hyp. and
7.ax.
C 33.6.
d 20. 3.
e 1. ax. 1.
f 32. I.
g 4. 6.
h 17.6.
k 27.3.
m 4. 1.
n 27.3.
O 32. I.
P 4. 6.
917.6.
I 2. 2.
£ 2. axe
Draw
234
The thirteenth Book of
a 6.2.
b confr.
€ 47. I.
d 3. ax.
e 17. 6.
£ 9. 13.
g 10. 13.
h47. I,
10.6.
a-cor. 10.
13.
b 32. 1.
€4.6.
d 15. 5-
e. 18.5.
f 22. 6.
gl. 13.
h 9. 10.
k 74. 10.
1 9. 10.
*
}
4 def. 85.
10.
m cor. 8.6.
and 17. 6.
n 95. 10.
Draw the diameter AB, to which erect a perpendicu-
lar CD at the center C, divide CB equally in E,
and make EFED, then DF fhall be the fide of the
pentagon.
For BF x FC+E€q (a)=EFq (b) EDq (c) =ECq+
ECq. (d) therefore BFX FC=DCq or ECq. (e) wherefore
BF: EC:: BC: FC; therefore fince BC is the fide of a
hexagon, (f) FC fhall be the fide of a decagon. Con-
fequently DF (b) = y DCq+FC (g) is the fide of a
pentagon. Which was to be done.
PROP. XI. Plate VII. Fig. 3.
If in a circle ABCD, whofe diameter AC, is rational, en
equilateral pentagon be infcribed ABCDF; the fide of the
pentagon AB is an irrational-line of that kind which is
called a minor-line.
Draw the diameter BFH, and the right-lines AC,
AH; and * make FL of the radius FH; and M
CA.
4
Becauſe the angles AKF, AIC, are ()right-angles, and
CAI common, the triangles AKF, AIC, are (b) equiangu-
lar: (c) therefore CI: FK (c): :CA: FA (FB) d): : CM:
FL; therefore by permutation FK: FL:: CI: CM (d)
:: CD: CK (2 CM) and fo by (e) compofition CD+CK
: CK:: KL: FL; (f) confequently Q: CDCK (g)
(5 CKq): CKq:: KLq: FLq. therefore KLq=5 FLq.
wherefore if BH (p) be taken 8, FH fhall be 4. FL 1, and
FLq 1, BL 5, and BLq 25, KLq 5. by which it appears
that BL and KL are § (4), (4) and fo BK is a refidual
and KL its congruent or adjoining-line; but fince BLq
KLq.—20.(/)
20.(/) thence BL BLq-KLq * whence
Therefore becauſe
BK fhall be a fourth refidual-line.
ABq (m) is HB × BK, (~) fhall AB be a minor-line.
Which was to be demonftrated.
PRO P. XII. Fig. 4.
If in a circle ABEC an equilateral triangle ABC be
infcribed, the fide of that triangle AB is in power tri-
ple to the line AD drawn from D the center of the
circle to the circumference,
The
*^
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Plate VI. Facing Pag. 234.
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​>
EUCLIDE's Elements.
235
The diameter being extended to E, draw BE. Bc.
caufe the arch BE (a) EC, the arch BE is the fixth
part of the circumference, (b) therefore
hence AEq (c) 4 DEq (4 BEq) (d) =
+ADq.) (e) confequently ABq3 ADq.
sole dem,
BE — DE.
ABq + BEq
ABq + BEq
Which was
a cor. 10.
13.
bior. 15.4.
€ 4. 2.
d 47°
47. I.
€ z. ax. 1.
Coroll.
1. AEq: Aßq: : 4! 3.
2. ABq: AFq:: 4:3•' (ƒ) For ABq : AFq: : AEq: _f cor. 8. 6.
ABq.
3. DFFE. For the triangle EBD (g) is equi-
lateral, (b) and BF perpendicular to ED: (b) there-
fore EF FD.
4. Hence, AF DE † DF3 DF.
PRO P. XIII. Plate VII. Fig. 5, 6.
To describe a pyramid EGFI, and comprehend it in a
phere given and to demonflrate that the diameters of the
Sphere AB is in power fefquialter of the fide EF of the
įyramid EGFI.
About AB defcribe the femicircle ADB ; (a) and let
AC be 2 CB. From the point C erect the perpendi-
cular CD, and join AD, DB, then at the interval of the
radius HE CD defcribe the circle HEFG, (b) wherein
infcribe the equilateral triangle EFG ; from H(c) erect IH
CA perpendicular to the plane EFG, produce IH to
K, (d) fo that IK AB; and join the right-lines IE, IF,
IG. Then EFGI fhall be the pyramid requried.
For becauſe the angles ACD, IHE, IHF, IHG, (e) are
right-angles; and CD, HE, HF, HG (e) equal, (e) and
IHAČ; (f) therefore AD, IE, IF, IG, fhall be equal
among themfelves, But becanfe AC (2 CB,): CB (g)::
ACq: CDq. thence fhall ACq 2 CDq; therefore ADq
(f) ACq+CDq (b) = 3 CDq≈ 3 HEq (k)=FFq ;
(1) therefore AD, EF, IE, IF, IG are equal, and ſo the
pyramid EFGI is equilateral. But if the point C be placed
upon H, and AC upon HI, the right-lines AB, IK, (m)
fhall agree, as being equal. Wherefore the femicircle
ABD being drawn about the axis AB or IK (2
fhall pass by the points E, F, G, * and fo the pyramid EF-
GI ſhall be inſcribed in a ſphere. Which was to be done.
Alfo
and 22. 6.
gcor 15.4.
h cor. 3. 3.
a 10. 6.
bcor. 15.4.
C 12. II.
d 3.
dz. 1.
e conftr.
f 4. 1.
g 20. 6.
h 2. ax.
k 12. 135
11.ax. 1.
m 8. ax. 1.
n15. def. i
31. def.
*
236
The thirteenth Book of
O
cor. 8. 6.
P confir.
Alſo it is manifeft that BAq: ADq(0)
: : 3:2.
:
BA: AC(A)
AC(†)
Which was to be demonftrated.
Coroll
q 12. 13.
& confir.
a 46. 1.
b 12. 11.
C 3. I.
d4. 1.
e 27. def.
II.
confir.
847.1.
:
=
1. ABq: HEq: 9:2. For if ABq be put 9,
then ADq (EFq) fhall be 9. (9) confequently HEq fhal!
be 2.
I.
(r)
Hence,
2. If L be the center, then fhall AB: LC: : 6:
For if AB be put = 6. then AL fhall be 3;
and thence AC 4. wherefore LC fhall be 1.
3. AB: HI::6:4:: :32.
: 2. whence.
4. ABq: HIq: 9:4.
PRO P. XIV. Plate VII. Fig. 7, 8.
To defcribe an Otaedron KEFGDL, and comprehend it
in the given Sphere, wherein a pyramid is: and to
and to de.
monftrate that AH, the diameter of the sphere, is in
power double of AC, the fide of that O&taedron.
About AH defcribe the femicircle ACH, and from
the center B erect the perpendicular BC, draw AC, HC;
then upon ED = AC (a) make the fquare EFGD, whofe
diameters DF, EG, cut in the center I; from I, draw
IL AB (b) perpendicular to the plane EFGD, produce
IL (c) till IK = IL. and join KE, KF, KG, KD, LE,
LF, LG, LD; then fhall KEFGDL be the Octaedron
required.
For AB, BH, FI, IE, & being femidiameters of
equal fquares are equal one to the other; (d) whence the
bafes LF, LE, FE, &c. of the right-angled triangles
LIE, LIF, FIE,&c. are equal, and confequently the eight
triangles LFE, LFG, LGD, LDE, KFF, KFG, KGD,
KDE, are equilateral, (e) and make an Octaedron, which
may be infcribed in a ſphere, whofe center is I, and
IL or AB the radius, (becauſe AB, IL, IF, IK, &c.
(f) are equal.) Which was to be done. Moreover, it
is evident that AHq (LKq) (g) = 2 ACq (2 LDq.)
Which was to be demon.
Coroll
1. Hence it is manife, that in the Octaedron the
shree diameters EG, FD, LK cut one the other por-
pendicularly in the center of the fphere.
$
2. Alle
EUCLIDE's Elements.
237
2. Alſo, that the three planes EFGD, LEKG, LFKD
are fquares, cutting one another perpendicularly.
3. The Octaedron is divided into two like and equal
pyramids EFGDL, and EFGDK, whoſe common baſe
is the fquare EFGD.
4. Laftly, it follows (b) that the oppofite bafes of the
Octaedron are parallel one to the other.
PROP. XV. Plate VII. Fig. 9, 10.
To defcribe a cube EFGHIKLM, and comprehend it
in the fame Sphere, wherein the former figures were; and
to demonftrate that AB the diameter of the sphere is in
power triple to EF, the fide of that cube.
Upon AB defcribe a femicircle ACB; (a) and make
AB 3 DA, and from D raiſe the perpendicular DC,
and join BC and AC. Then upon EFAC (6) make
the fquare EFGH, upon whofe plane let the right-lines
EI, FK, HM, GL, ftand perpendicular, being equal to
EF, and connect them with the right-lines IK, KL, LM,
IM.
The folid EFGHIKLM, is a cube, as is fuffici-
ently apparent from the conftruction.
In the oppofite fquares EFKI, HGLM, draw the di-
ameters EK, FI, HL, MG, through which let the planes
EKLH, FIMG be drawn, cutting one another in the line
NO, which (c) fhall divide equally in two parts the diame-
ters of the cube EL, FM, GI, HK, in Pthe center of the
cube; (d) therefore P fhall be the center of a ſphere paffing
through the angular points of the cube. Moreover,
ELq (e) = EKq+KLq (e) = 3 KLq, (f) or 3 ACq.
but ABq: ACq (g): :BA: DA (ƒ): :3 : 1; (h) there-
fore AB EL ; wherefore we have made a cube, &c.
Which was to be done.
Coroll.
1. Hence it is manifeft that all the diameters of the
cube are equal one to the other, and do equally biſect one
another in the center of the ſphere. And for the fame
reaſon the right-lines which conjoin the center of the
oppofite fquares are bifected in the fame center.
2. The diameter of a ſphere containeth in power the
fide of a tetraedron and of a cube, viz. ABq (k) ≈ (/)
BCq † (22) ACq.
h 15. 11.
a 10. 6.
b 46. 1.
c cor. 39.
II.
d 15. def.
1. and 14.
def. 11.
e 47. 1.
f confir.
gcor. 8. 6.
h 14.5-
k 47. £.
1 13. 13-
m 15. 13.
PROP.
238
The thirteenth Book of
a 10. 6.
bil.4.
C 12. II.
d conftr.
e 6.11.
£ 33. 1.
g15.II.
h 1. def. 3.
k 47. 1.
1 confir.
m to. 12.
n fch. 48. I.
and ax. 1.
o cor. 14.
II.
P 47. I.
q 10. 13.
PRO P. XVI. Plate VII. Fig. 11, 12.
1
To deſcribe an Icosaedron ZGHIK FYVXRST, and
encompass it in the fphere, wherein were contained, the
forefaid folids; and to demonstrate that FG the file of
the Icofaedron is that irrational-line, which is called a
minor-line.
C.
Upon AB the diameter of a fphere defcribe the
femicircle ADB ; and (a) make AB equal 5 BC, then
from C erect CD perpendicular, and draw AD and BD.
At the diſtance EF BD defcribe the circle EFKNG;
(b) wherein infcribe the equilateral pentagon FKIHG.
Divide equally in two parts the arches FG, GH,
&c. and join the right-lines FL, LG, &c. being the
fides of a decagon. Then (c) erect EQ, LR, MS, NT,,
OV, PX equal to EF, and perpendicular to the plane
FKNG; and connect RS, ST, TV, VX, XR; as alfö
FX, FR, GR, GS, HS, HT, IT, IV, KV, KX.
Laftly, produce EQ, and take QY FL, and EZ-
FL, and conceive the right-lines ZG, ZH, ZI, ZK,
ZF to be drawn ; as alfo YV, YX, YR, YS, YT.
Then I fay the Icofaedron required is made.
For becauſe EQ, LR, MS, NT, OV, FX, are (d)
equal and (e) parallel, alfo thoſe lines that join them
EL, QR, EM, QS, EN, QT, EO, QV, EP, QX,
(f) are equal and parallel. And thence likewife LM
(or FG) RS, MN, ST, &c. are equal one to the
other; (g) therefore the plane drawn through EL,
EM, &c. is equidiftant from the plane paffing through
QR, QS, &c. (b) and the circle QXRSTV drawn
from the center Qis equal to the circle EPLMNO,
and RSTVX is an equilateral pentagon. But EF, EG,
EH, &c. and QX, QR, QS, &c. being conceived to
be drawn ; then becauſe FRq (k) = FLq + LRq, (!)
or EFq (m) = FGq. (2) therefore FR, FG, and fo all
RS, FG, FR, RG, GS, GH, &c. fhall be equal one
to the other, and confequently the ten triangles RFX,
KFG, RGS, &c. are equilateral and equal. More-
over, becauſe XQY is a (0) right-angle; therefore
XYq (p)= QXq + QYq (q) = VXq or FGq; where-
fore XY, VX, and likewife YV, YT, YS, YR, ZG,
ZH, &c. are equal. Therefore other ten triangles are
made, equilateral and equal both to one another,
and to the ten former, and fo an Icofaedron is
made
$
Moreover
EUCLIDE's Elements.
239
Moreover, divide equally EQ in a, draw the right
lines, a F, a X, a V ; and becauſe QX (r) =QV, and
a the common fide, and EQX, EQV are right-an-
gles (therefore fhall a X be aV; and for the fame
reafon all the lines a X, aR, a S, a T, a V, a Ề, a G,
a H, a I, a K are equal. But becauſe ZQ: QE (t):
QEZE, therefore Zaq (u) = 5 Eaq (x)=EQg (EFq)
+Eaq (y) aFq; therefore ZaaF; in like man-
ner af Ya; therefore the fphere, whofe center is a
and aF the radius, fhall pass through the 12 angular
points of the Icosaedron.
Laftly, (2) becaufe Za: aE:: ZY: QE; (a) and fo
ZaqaEq:: ZYq: QEq. (b) therefore ZYq = 5 QEq,
or 5 BDq; but ABq: BDq (c): : AB: BC: 5:5 (4)
therefore ZY AB, Which was to be done. Therefore
if AB be put §, (e) then EF = √ AB× BC ſhall be
alfo p, and confequently FG the fide of the pentagon,
and likewife of the Icofaedron, (f) is a minor-line.
I'hich was to be dem.
1
r15. def. 1.
£ 4. 1.
t 9. 13.
u 3. 13.
X 4. 2.
Ꭹ 47, 1.
z 15. 5.
a 22.6.
b 14. 5.
c cor. 8. 6..
d 1. ax. I.
eſch. 12.
IO.
f II. 13.
+
Coroll.
1. From hence is inferred, that the diameter of
the fphere is in power quintuple of the femidiamater
of the circle encompaffing the five fides of the Ico-
faedron.
2. Alfo it is manifeft that the diameter of the
fphere is compofed of the fide of a hexagon, that is,
of the femidiameter, and two fides of the decagon of
a circle encompaffing the five fides of the Icofaedron.
3. It appears likewife that the oppofite fides of an
Icofaedron, fuch as RX, HI, are parallels. For RX
(a) is parallel to LP, (b) parallel to HI.
·
PRO P. XVII. Plate VII. Fig. 13.
To defcribe a Dodecaedron, and comprehend it in the ſphere
wherein the former figures were comprehended; and to de-
monſtrate that the fide RS of the Dodecaedron is an irratio-
nal-line of that fort which is called an apotome or reſidual-
line.
Let AB be a cube inſcribed in the given ſphere, and
let all the fides thereof be divided equally in the points
E, H, F, G, K, L, &c. and join the right-lines KL, MH,
HG, EF; (a) make HI: IQ: IQ: QH; and take NO,
NP,
a 33. 1.
b ſch. 26.
3.
a 30.6.
240
The thirteenth Book of
a 47. 1.
b 7. ax.1.
C 4. 13.
d 47. I.
€ 4. 2.
f conftr. 96.
11.
g 33. 1.
hg. 1.
k 7.11.
k conftr.
16. 11.
m 32. 6.
n 1, and 2.
11.
05. 13.
P 47. 13.
q 1. ax. 6.
and 4. 13.
I 4. 2.
f 8. I.
*
7. 13.
t 15. 13.
u 1. ax. I.
X 29. I.
2 47. I.
a 4. 13.
b 15. 13.
NP, IQ; then erect OR, PS, perpendicular to the
plane DB, and QT to the plane AC; and let OR, PS,
QT, be equal to IQ, NO, NP; whence DR, RS, SC,
CT, TD, being connected, DRSCT fhall be a pentagon
of the Dodecaedron required. For draw NV parallel to
OR, and having drawn NV out as far as the center of the
cube X, join the right-lines DS, DO, DP, CR, CP, HV,
HT, RX. Becauſe DOq (a) — DKq (6) (KNq)+KOq
(c) = 3 ONq (3 ORq) (a) thence DRq= 4 ORq (e) =
OPq, or, RSq. therefore DR=RS. By a like way of
reafoning DR, RS, SC, CT, TP are equal. But becauſe
OR () is equal and (g) parallel to PS, therefore. RS,
OP, and confequently RS, DC fhall be alfo parallels;
(b) therefore theſe with thoſe that conjoin them DR,
CS, VH, are in one and the fame plane. More-
over, becauſe HI : IQ (k) : : IQ (TQ): QH (k) : : HN
: NV; and both TQ, HN, and QH, NV (k) are per-
pendicular to the fame plane, () and fo alfo parallels,(m)
THV ſhall be a right-line; (2) therefore the Trapezium
DRSC, and the triangle DTS are in one plane ex-
tended through the right-lines DC, TV; (0) therefore
DTCSR is a pentagon, and that alfo equilateral, by
what is fhewn already. Furthermore, becauſe PK: KN
:: KN: NP; and DSq (p) DPq+ PSq (PNq)
(p) DKq+PKq+NPq, (2) thence DSq=DKq+
3 KNq4 DKq (4 DHq) () DCq; therefore DS
DC; whence the triangles DRS, LCT, are equila-
teral one to another; () therefore the angle DRS=
DTC, and likewife the angle CSRDCT; therefore
the * pentagon DTCSR is alfo equiangular. Moreover,
becaufe AX, DX, CX, &c. are femidiameters of the
cube (t) thence is XN=IH, or KN, (2) and fo XV
KP; wherefore becauſe RVX, is a (x) right-angle, (x)
thence RXq=XVq+RVq (NPq)=Kq+NPq (a)
== =
3 KNq (6) AXq or DXq, &c therefore RX, AX,
=
DX, and for the fame reaſon XS, X'T, AX, are equal
one to another And if by the fame method whereby
the pentagon DTCSR was made, twelve like penta-
gons, touching the twelve fides of the cube, be made,
they ſhall compoſe a Dodecaedron; and a ſphere paffing
through their angular points, whofe radius is AX, or
RX, fhall comprehend that Dodecaedron. Which was
to be done.
Laſtly, becauſe KN: NO (c):: NO: OK; (d) thence
KL: OP::OP: OK+PL. Therefore if AB the diameter
of
EUCLIDE'S Elements.
245
of the fphere be fuppofed p, then ſhall KL (e)
AB
y
3
(f) be alfo p. (g) whence OP, or RS the fide of the Do-
decaedron fhall be a refidual-line. Which was to be dem.
Coroll.
From this demonſtration it follows, 1. That if the fide
of a cube be cut in extreme and mean proportion, the
greater fegment fhall be the fide of the Dodecaedron de-
fcribed in the fame ſphere.
2. If the leffer fegment of a right-line, cut in extreme
and mean proportion, be the fide of the Dodecaedron,
the greater ſegment fhall be the fide of the cube infcribed
in the fame fphere.
3. It is manifeft alfo, that the fide of the cube is equal
to the right-line which fubtendeth the angle of a penta-
gon of the Dodecaedron infcribed in the fame ſphere.
PRO P. XVIII. Plate VII. Fig. 14.
To find out the fides of the five precedent figures, and com-
pare them together.
Let AB be the diamter of the fphere given, and AEB
the femicircle, and let AC be (a) = AB, and AD (6)
AB; then erect the perpendiculars CE, DF, and
BG=AB; join AF, AE, BE, BF, CG; and let fall
the perpendicular HI from H, and CK being taken equal
to CI, from K erect the perpendicular KL, and join AL.
Laftly, (c) make AF: AO AO : OF.
Therefore 3 : 2 (d) : : AB : BD (e) :: ABq: BFq the
fide of a Tetraedron, and 2 : 1 :: (a) AB: AC :: ABq:
BEq (f) the fide of an Octaedron.
Alfo 3: 1(d):: AB: AD (e) :: ABq: AFq. (g) the
fide of an Hexaedron.
Moreover, becauſe AF AO (b): : ÁO: OF ; (k);
thence fhall AO be the fide of a Dodecaedron. Laftly,
BG (2 BC) : BC (/) : : HI: IC; (m) therefore HI2
CI (KI; therefore HIq (0) 4 CIq. confequently
CHq) 5 CIq; (4) therefore ABq5 KIq. (7)
therefore KI, or HI is a radius of a circle encloſing the
pentagon of an Icofaedron; and AK or IB (r) is the
fide of a decagon inſcribed in the fame circle ; (/)
whence AL fhall be the fide of a pentagon, (t) and alio
the fide of an Icofacdron. Whereby it appears that BF,
Q
BE,
e 15. 3.
£ Schol. 12.
.10.
g 6. 13.
a io. i.
b 10. 6:
c 30.6.
d conftr:
e cor. 8.6f
f 14. 13.
I
g 15. 13:
h confir
k cor. 17:
13.
1 4. 6.
m 14. 5.
n conftr.
04. 2.
P 47. I.
q 15. 5.
I cor. 16.
13.
f 10: 13:
¿ 16. 13:
242
The thirteenth Book of
u 1.6.
x 4. ax. 1.
y I. 2.
z 17.6.
a 47. I.
a 21.11.
b See Sch.
32. I.
BE,AF are and AL, AO and BFBE, and
BE AF, and AF AO. And becauſe 3 AFq=
ABq (u)= 5 KLq, and AF x AO AFX OF, (x) and
fo AF × AO↓ AF× OF 2 AF × OF, (y) that is,
AFq (≈) 2 AOq; (a) thence fhall 3 AFq (5 KLq)
be 6 A0q. confequently KL AO, and much ra-
ther ALC AO.
That we may expreſs theſe fides in numbers; If AB
be ſuppoſed ✔✅ 60, then, reducing what is already fhewn
to fupputation, BF 40, and BE 30, and AF
=√
√302
20. Alfo AL√ 30-180 (for AK=√ 15
--3. and KL (HI) 12:) Laftly, AO: 30-
✔ 500 (√ 25 — √√ 5.)
Schol.
It is very apparent that befides the five aforefaid figures,
there cannot be described any other regular folid figure (viz.
fuch as may be contained under ordinate and equal plane
figures.)
For three plane angles at leaſt are required to the con-
ftituting of a folid angle: (a) all which must be lefs than
four right-angles. (b) But 6 angles of an equilateral tri-
angle, 4 of a fquare, and 6 of a hexagon, feverally
equal 4 right-angles; and 4 of a pentagon, 3 of a hep-
tagon, 3 of an octagon, &c. exceed 4 right-angles:
Therefore only of 3, 4, or 5 equilateral triangles, of 3
fquares, or 3 pentagons, it is poffible to make a folid
angle. Wherefore befides the five above mentioned,
there cannot be any other regular bodies.
Out of P. Herigon.
The Proportions of the sphere and the five regular figures
infcribed in the fame.
Let the diameter of the fphere be 2, then fhall the
periphery or circumference of the greater circle,
be 6. 28318.
The fuperficies of the greater circle, 3. 14159.
The fuperficies of the ſphere, 12. 56637.
The folidity of the fphere, 4- 1879.
The fide of the tetraedron, 1. 62299.
The fuperficies of the tetraedron, 4. 6188.
The folidity of the tetraedron, o. 15132.
The
EUCLIDE'S Elements.
243
The fide of the hexaedron, 1. 1547:
The fuperficies of the hexaedron, 8.
The folidity of the hexaedron, 1. 5396.
The fide of the octaedron, 1. 41421.
The fuperficies of the octaedron, 6. 9282.
The folidity of the octaedron, 1. 33333.
The fide of the dodecaedron, o. 71364.
The fuperficies of the dodecaedron, 10. 51462.
The folidity of the dodecaedron, 2. 78516.
The fide of the Icofaedron, 1. 05146.
The fuperficies of the Icofaedron, 9. 57454
The folidity of the Icofaedron, 2. 53615.
The End of the Thirteenth Book.
THE
Q_2
[244]
The FOURTEENTH Book
O F
EUCLIDE's
ELEMENT S.
a 4. I.
b
5. I.
€ 32. 1.
d byp. and
33.6.
e 20. 3.
£7. ax.
f
EA
6. 1.
PROPOSITION I. Plate VII. Fig.15.
A
Perpendicular-line DF drawn from D the center of
a circle ABC to BC the fide of a pentagon infcri-
bed in the faid circle, is the half of these two lines
taken together, viz. of the fide of the hexagon DE, and the
fide of the decagon EC infcribed in the fame circle ABC.
Take FG FE, and draw CG; (a) Then CE is≈
CG. Therefore the angle CGE (b)=CEG (b) =ECD;
therefore the angle ECG (c)= EDC (d) = ADC (6)
= CED (ECD;) (f) confequently the angle GCD
= ECG = EDC,
(g) wherefore DG = GC (CE. )
DE CE.
therefore DFCE (DG) +EF=
avas to be demonfirated.
2
Which
a 17.6.
b 8.2.
CI. ax. I.
322 5.
and 22. 6.
e 22.5.
£ 7. 5.
AGB C
D H E
PROP. II.
If two right-lines AB, DE, are
cut according to extreme and mean
F proportion (AB: AG:: AG : GB.
and DE: DH:: DH: HE.) they
fhall be cut after the fame manner,
viz. in the fame proportions (AG: GB:: DH: HE.)
Take BC BG; and EF = EH.
- EH. Then ABX BG is
(a) AGq. wherefore ACq (6) 4 ABG-AGq (c) = 5
AGq. In like manner fhall DFq=5 DHq. (d) therefore
AC AG: DF: DH. whence by compounding AC-
AG: AG :: DF + DH: DH. that is, 2 AB: AG: : 2
DE : DH, (e) confequently AB: AG :: DE: DH; (ƒ)
whence by diviſion AG: GB :: DH: HE. Which was
to be demonfirated.
•
PROP.
EUCLIDE's Elements.
245
PRO P. III. Plate VII. Fig. 16, 17.
The fame circle ABD comprehends both ABCDE the pen-
tagon of a Dodecaedron, and LMN the triangle of an
Icofaedron infcribed in the ſame ſphere.
afch.47.1.
b
30.6.
C 47. I.
d
4. 2.
e 10. 13.
ميع
3 ax.
g 8. 13.
2. 13. &
16.5.
k 22.6.&
Draw the diameter AG, and the right-lines AC, CG;
and let IK be the diameter of the ſphere, (a) and IKq
5 OPq. (b) and make OP: OQ:: OQ: QP. Becaufe
ACq+CGq (c) AGq (d)=4FGQ; and ABq (e)= f 2. and
FGq+CGq. (f) thence ACq+ABq5 FGq; more-
over, becauſe CA : AB (g) : : AB: CA-AB; and OP :
OQ:: OQ: QP; (b) and as CA: OP: AB: OQ (4)
therefore 3 ACq (4) (IKq.): 5 OPq (m) (IKq): : 3 ABq :
50Qq. therefore 3 ABq5 OQq. But becauſe ML
(") is the fide of a pentagon infcribed in a circle, whoſe
radius is OP, thence 15 RMq (0) = 5 MLq (p)
OPq + 500 q = * 3 Acq+3 ABq (9)= 15 FGq; (") m confir.
therefore RMFG. () and confequently the circle
ABD is the circle LMN. Which was to be demonſtrated.
PROP. IV. Fig. 18, 19.
If from F the center of a circle encompaſſing the penta-
gon of a Dodecaedron ABCDE, a perpendicular-line FG be
drawn to one fide of the Pentagon CD; the rectangle con-
tained under the faid fide CD and the perpendicular FG,
being thirty times taken, is equal to the fuperficies of a
Dodecaedron. Also,
If from the center L of a circle inclosing the triangle of
an Icofaedron HIK, a perpendicular-line LM be drawn to
one fide of the triangle HK, the rectangle contained under
the faid fide HK and the perpendicular LM, being thirty
times taken, fhall be equal to the fuperficies of an Icofaedron.
Draw FA, FB, FC, FD, FE; (a) then ſhall the tri-
gles CFD, DFE, EFA, AFB, BFC be equal, but CDx
FG (6) 2 triangles CFD; therefore 30 CDX GF (c)=
60 CFD (d) = 12 pentagons ABCDE (e) to the fuper-
ficies of a Dodecaedron. Which was to be demonftrated.
Draw LI, LH, LK; then HK × LM (ƒ) = 2 triangles
LHK; therefore 30 HK X LM (g) 60 HLK
HIK (b)
the fuperficies of an Icofaedron. Which was
==
to be demonftrated.
Q 3
20
Coroll
4. 5.
1 15. 13.
n cor. 16.
13.
0 12. 13.
p 10. 13.
9 15.5.
and above
* before
r 1. ax. I.
and fchol.
48. I.
11. def. 3.
a 8. 1.
b 41. 1.
d 6. ax.
c 15.5.
e 17. 3.
41. I.
f
g 15. 5.
h16.13.
246
The fourteenth Book of
★ 15. 5. ]
23. 14.
b 9. 13.
C 1, 14:
d cor. 12,
13.
e 15.5.
£ cor, 179
13.
g 2. 14.
1. 6.
k
hi.
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1 cor. 4. 14:
a cor, 17.
13.
b 12. 13.
£ 4. 13.
d 15.5.
e 2. 14.
£72. 6.
Coroll.
CD XFG: HK X LM (): the fuperficies of a Dode-
caedron to the fuperficies of an Icofaedron.
PROP. V. Plate VII. Fig. 20.
The fuperficies of a Dodecaedron hath to the fuperficies of an
Icofaedron inſcribed in the ſame ſphere, the fame proportion
that H the fide of a cube hath to AD the fide of an
Icofaedron.
Let the circle ABCD (a) encloſe both the Pentagon of
a Dodecaedron, and the triangle of an Icofaedron; whoſe
fides are BD, AD; upon which from the center E let
fall the perpendiculars 'EF, EGC; and draw CD.
Becauſe EC+CD : EC (b) : : EC: CD. thence EG
(c) ({ ECX / CD) : EF (d) (½ EC) (e) : : EF : EG-EF
(CD.) but H: BD (ƒ) : : BD : H-BD; (g) therefore
H: BD :: EG: EF. confequently H x EF BD× EG.
wherefore fince H: AD (b):: HX EF: AD X FF; (k)
thence fhall be H : AD: : BD × EG : AD × EF (1)
the fuperficies of a Dodecaedron to the fuperficies of an
Icofaedron. Which was to be demonftrated.
PROP. VI. Fig. 2 21.
If a right-line AB be cut in extreme and mean proportion,
then as the right-line BF, containing in power that which is
made of the whole line AB, and that "which is made of the
greater fegment AC, is to the right -line E containing in pow-
er that which is made of the whole line AB, and that which
is made of the leffer fegment BC; fo is the fide of the cube
BG to the fide of an Icofaedron BK infcribed in the fame
Sphere with the cube.
In the circle, whofe femidiameter is AB, infcribe
BFGHI the pentagon of a Dodecaedron; and BKL the
triangle of an Icofaedron, (a) wherefore BG fhall be the
fide of a cube infcribed in the fame fphere; therefore
BKq(b) 3 ABq; and Eq (c) 3 ACq. therefore BKq:
=
Eq (d): ABq: ACq (e) : : BGq: BFq; wherefore by
permution BGq: BKq:: BFq: Eq; (f) whence BG:
BK: BF E. Which was to be demonstrated.
PROP.
EUCLIDx's Elements.
247
PROP. VII.
A Dodecaedron is to an Icofaedron, as the fide of a Cube
is to the fide of an Icofaedron, infcribed in one and the
Same Sphere.
Becauſe (a) the fame circle comprehends both the pen-
tagon of a Dodecaedron, and the triangle of an Icofae-
dron, (b) the perpendiculars drawn from the center of the
ſphere to the planes of the pentagon and triangle, ſhall
be equal one to another. Therefore if the Dodecaedron
end Icofaedron be conceived and divided into pyramids,
right-lines being drawn from the center of the ſphere
to all the angles, the altitudes of all the pyramids ſhall
be equal one to the other. Wherefore fince the pyra-
mids are (c) of equal height with the bafes, and the fu-
perficies of the Dodecaedron is equal to twelve penta-
gons, and the fuperficies of the Icofaedron to twenty
triangles, the Dodecaedron ſhall be to the Icofaedron, as
the fuperficies of the Dodecaedron is to the fuperficies of
the Icofaedron; (d) that is, as the fide of the cube is to
the fide of the Icofaedron.
PRO P. VIII. Plate VII. Fig. 22, 23.
The fame circle BCDE comprehends both the Square
of the cube BCDE; and the triangle of the o&taedron FGH
infcribed in one and the ſame ſphere.
Let A be the diameter of the ſphere. Becauſe Aq (a)
3 BCq (b) = 6 BIq; and alfo Aq (c) = 2 GFq (d)
6 KFq; thence ſhall BI=KF; (e) therefore the circle
CBED=GFH. Which was to be demonftrated.
a 3. 14.
b 47. 1.
c 5. and 6:
12.
d 5. 14.
a 15. 13.
b 47. 1.
C 14. 13.
d 12. 13.
e 1. def. 3.
The End of the Fourteenth Boox.
Q4
THE
(248)
7
The FIFTEENTH BOOK
O F
EUCLIDE's
ELEMENTS.
2 47. 1.
}
b 31. def.
ΙΙ.
a 10. I.
b 4. I.
€ 27. def.
II.
₫ 31. def.
RROPOSITION I. Plate VII. Fig. 24.
I
Na cube given ABGHDCFE to defcribe a pyramid
AGEC.
From the angle C draw the diameters CA, CG,
CE; and connect them with the diameters AG, GE, EA,
All which are (a) equal among themſelves, as being the
diameters of equal fquares: therefore the triangles CAG,
CGE, CEA, EAG are equilateral and equal; and con-
fequently AGEC is a pyramid, which infifts upon the
angles of the cube, and therefore (b) is infcribed in it.
Which was to be done.
f
PROP. II. Fig. 25.
In a pyramid given ABDC to defcribe an Octaedron
EGKIFH.
;
(a) Bifect the fides of the pyramid in the points E,
I, F, K, G, H, which join with the 12 right-lines EF,
FG, GE, &c. All theſe are (b) equal one to the other
confequently the 8 triangles EHI, IHK, &c. are equi-
lateral and equal, and fo make (c) an Octaedron defcri
bed (a) in the given pyramid. Which was to be done.
PROB
EUCLIDE's Elements.
249
PROP. III. Plate VII. Fig. 26.
In a cube given CHGBDEFA to defcribe an Octaedron
NPQSOR.
* 8, 4•
a 4. I.
Connect * the centers of the fquares N, P, Q, S, O, R
with the twelve right-lines NP, PQ, QS, &c. which
are (a) equal among themfelyes; and fo make 8 equila-
teral and equal triangles: wherefore (b) the Octaedron
NPQSOR (b) is inſcribed in the cube. Which was to be 27. def. 11.
done.
PROP. IV. Fig. 37.
1
In an Octaedron given ABCDEF, to infcribe a cube.
Let the fides of the pyramid EABCD, whoſe baſe
is the fquare ABCD, be bifected by the right-lines,
LM, MÑ, NO, OL, which are (a) equal and (b) pa-
rallel to the fides of the fquare ABCD; (e) then
the quadrilateral LMNO is a fquare. In like manner,
if the fides of the fquare LMNO be bifected in the
points G, H, K, I, and GH, HK, KI, IG connected,
GHKI fhall be a fquare. And if in the other
pyramids of the Octaedron, the centers of the triangles
be in the fame fort conjoined with right-lines, then
other fquares will be defcribed like and equal to the
fquare GHKI; wherefore fix fuch fquares fhall make
a cube, which fhall be defcribed within an Octaedron,
(d) fince its eight angles touch the eight bafes of the
Octaedron in their centers. Which was to be done.
PRO P. V. Fig. 28.
5
In an Icofaedron given to infcribe a Dodecaedron.
Let ABCDEF be a pyramid of the Icofaedron, whoſe
bafe is the pentagon ABCDE; and the centers of the
triangles G, H, I, K, L; which connect with the right-
lines GH, HI, IK, KL, LG. Then GHIKL fhall be
a pentagon of the Dodecaedron to be infcribed.
For the right-lines, FM, FN, FO, FP, FQ, paffing
through the centers of the triangles, (a) bifect their bafes;
(b) therefore the right-lines MN, NÓ, OP, PQ, QM,
(c) are equal one to the other; (d) whence alfo the angles
MEN
b
31. and
a 4. I.
b 2. 6.
c29.def.1.
d 31. def.
Ii.
a cor. 3. 3•
b 4. 1.
C 4. I.
ds. i
250
The Fifteenth Book of
€ 4. t.
f 12. 13.
MFN, NFO, OFP, PFQ, QFM are equal; therefore
the pentagon GHIKL is equiangular,(e) and confequently
equilateral, fince FG, FH, FI, FK, FL (f)are equal. And
if in the other eleven pyramids of the Ifocaedron, the
centers of the triangles be in like fort joined with right-
lines, then will the pentagons be equal and like to the
pentagon GHIKL be defcribed. Wherefore 12 of fuch
pentagons fhall conftitute a Dodecaedron; which alſo
hall be deſcribed in the Icofaedron, feeing the twenty
angles of the Dodecaedron confift upon the centers of
the twenty bafes of the Icofaedron. Whereby it ap-
pears that we have deſcribed a Dodecaedron in an Icola-
edron given. Which was to be done.
FINI S.
(251)
EUCLIDE's DATA.
DEFINITIONS.
"P"
Lanes or Spaces, Lines and Angèls, to which ave
can find others equal, are ſaid to be given in
Magnitude.
II. A Ratio is faid to be given, when we
or one equal to it.
can find it,
III. Rectiline-figures, whofe angles are given, and alfe
the ratio of the fides to one another, are said to be
given in Species or Kind.
IV. Points, Lines and Angles, which have and keep always
one and the fame place and fituation, are faid to be
given in Pofition or Situation.
V. A Circle is faid to be given in Magnitude, when the
femidiameter thereof is given in Magnitude.
VI. A circle is faid to be given in Pofition, and Mag-
nitude, when the center thereof is given in Pofition,
and the femidiameter in Magnitude.
VII. Segments of Circles, whofe angles and bafes are gi-
ven in Magnitude, are faid to be given in Magnitude.
VIII, Segments of a Circle, whofe angles are given in
Magnitude, and the bafes of the fegments in Pofition
and Magnitude, are faid to be given in Pofition and
Magnitude.
IX. A Magnitude AB, is greater than another Magni-
tude C, by a given Magnitude
D
BD, when having taken away
A-
the given Magnitude DB, the reft
AD, is equal to the other Mag-
-B
C
nitude C.
X. A Magnitude AB, is less than another Magnitude C₂
by a given Magnitude BD, when
baving added thereto the given
Magnitude BD, the whole AD is
equal to the other Magnitude C.
B
A-
C
-D
XI,
252
EUCLIDE'S DATA.
XI. A Magnitude AB is faid it be greater than another
D C
A-
Magnitude CB, by a given Mag-
nitude AD, and in ratio, when
B taken from the fame Magnitude
the given Magnitude AĎ, the
reft DB, hath to the other Magnitude ČB, a given ratis.
XII. A Magnitude AB is faid to be less than another
ה
Magnitude BC, by a given
A
B
D
-B
Magnitude AD, and in ratio,
vihen the given magnitude
AD being added thereto, the
whole DB hath to the other Magnitude BC, a given
ratio.
XIII. A right-line is faid to be drawn down from a given ·
point, unto a right-line given in Pofition, the right line
being drawn in a given angle.
XIV. A right-line is faid to be drawn up from a given
point, to a right-line given in Pofition, the right-line
being drawn in a given-angle.
"
XV. A right-line is against another right-line in Pofi-
tion, when it is drawn parallel thereto through a given
point.
a 1. def.
A B C D
b7.5.
c 16. 5.
d2. def.
PROPOSITION I.
Tw
WO Magnitudes A and B being given,
the ratio they have to one another A to
B is also given.
Demonftration. For feeing that the Mag-
tude A is given, (a) we can find one equal
thereto, which let be C. Again, forafmuch
as the Magnitude B is given, we can alfo find one
equal to that, and let that be D. Therefore ſeeing
that A is equal to C, and B to D, as A is to C, (b)
fo is B to D, and by permutation, (c) as A fhall be to
B, fo C fhall be to D. Therefore (d) the ratio of A to
Bis given, for it is the fame ratio as of C to D, as we
have found, and which ought to be demonftrated.
PRO P. II,
If a given Magnitude A, hath to fome other Magnitude
B, a given ratic, that other Magnitude B, is alfa given
in Magritude,
Demon.
EUCLIDE'S DATA.
253
A B C D
Demonft. For feeing that A is given, we
can find one equal thereto, which let be
C: And forafmuch as the ratio of A to B,
is alfo given, we can find (a) one of the
fame. Let it be found, and let the ratio
be of C to D. Now feeing that as A is.
to B, ſo C is to D; and by permutation, as A is to
C, fo B is to D: But A is equal to C, therefore (6) B
ſhall be alfo equal to D. Therefore (r) the Magnitude
B is given, feeing that thereto there hath been found one
equal, to wit, D.
}
PROP. III..
If given magnitudes AB and BC, are com-
pounded, that magnitude AC, that is compound-
ed of them, fhall be also given.,
+
A D
B E
CÉ
Demonftr. For fecing that AB is given, we
can find one equal to it, which let be DE.
Again, feeing that BC is given we can alſo
find one equal to that, which let be EF.
Wherefore feeing that DE is equal to AB,
and EF is equal to BC, the whole AC (a) is equal to the
whole DF. Therefore AC is given, feeing that DF is
propofed equal thereto.
PROP. IV.
If from a given magnitude AB, there be ta-
ken away a given magnitude AC, the remain-
ing magnitude CB is alſo given.
A D
CF
E
Demonfir. Forafmuch as AB is given, we
can find one equal thereto, which let be DE.
Again, fecing that AC is given, we can alſo
find one equal to it, which let be DF. Seeing
then that the Magnitude AB is equal to the
Magnitude DE, and the Magnitude AC to the Magni-
tude DF; the reſt CB (a) ſhall be equal to the reft FE.
Wherefore CB is given, for to it there hath been found.
an equal, to wit, FE.
a 2. def.
b 14. 5.
c 1. def.
a 2. ax. Ia
a 3-ax. 1.
PROP.
254
EUCLIDE'S DATA.
a 2. def.
b 2. prop.
c4. prop.
d 1. prop.
€ 19. 5.
a 2. prop.
b 3. prop.
c 1. prop.
A D
CF
BE
PROP. V.
If a Magnitude AB, bath a given ratio to
Jome part thereof AC, it will have alſo a given
ratio to the part remaining CB.
Demonftr. Let DE be expofed as a given
Magnitude, and feeing that the ratio of the
Magnitude AB, to Magnitude AC, is given
(a) we can find one of the fame, which let be
DE to DF; therefore the ratio of the fame
DE to DF is given. But DE being given, fo is (b) alſo
its part DF; and confequently, (c) the reft FE: Therefore
(d) feeing that DE and FE are given, the ratio of the
fame DE to FE is alfo given. And forafmuch as
DE is (e) to DF, as AB is to AC, and by converfion, as
DE to FE, fo AB is to CB. But the ratio of DE to
FE, is given, as hath been demonſtrated; therefore the
ratio of AB to CB is alſo given.
Scholium.
From this it is evident that if a Magnitude hath to
fome part thereof a given ratio, by divifion, the ratio
that one part hath to the other, fhall be alfo given.
For feeing that as DE is to FE, fo is AB to CB; by
divifion, as DF to FE, fo AC to CB. But it hath
been demonſtrated that the parts DF and EF are given,
and confequently their ratio is alſo given: In like man-
ner, therefore, the ratio of AC to CB is given.
A D
PRO P. VI.
If tawo Magnitudes AB and BC, having to
one another a given ratio, are compounded, the
Magnitude AC compounded of them, fhall also
BE have a given ratio to each of them AB and BC.
Demonftr. Let the given Magnitude DE be
propofed, and feeing that the reafon of AB to
F BC is given; let there be made one and the
fame of the faid DE to EF; therefore the ratio
of the fame DE to EF is given; and therefore (a) the
Magnitude DE being given, both the one and the other
of them DE and FE, is given. Wherefore (b) the whole
DF fhall be alfo given. Therefore (c) the raito of the
fame DF to each of them DE and FF, fhall be given.
And
EUCLIDE'S DATA.
255
1
And foraſmuch then as AB is to BC, ſo is DE to EF
by compounding, (d) as AC is to BC, fo is DF to EF: d 18. 5.
Therefore by converfion, as AC to AB, fo is DF to
DE. Therefore as the whole DF is to each of the other
Magnitudes DE and FE, fo the whole AC is to each of
the Magnitudes AB and BC. Therefore (e) the ratio of the 2 def.
the fame AC to each of the Magnitudes AB and BC is
given.
PROP. VII.
If a given Magnitude AB be divided
according to a given ratio AC to CB, A.
each fegment AC and CB is given.
C
-B
Demonftr. For feeing the ratio of AC to CB is given,
the ratio of (a) AB to each of them (AC and CB) is
alfo given. But AB is given: Therefore (6) each of the
fegments AC and CB is alſo given.
a 6. prop.
b2. prop.
PROP. VIII.
Magnitudes A and C, which have
to one and the fame a given ratio, B,
Shall be to one another in a given
ratio A to C.
Demonftr. For let the given mag-
nitude Ď be propofed, and feeing
that the ratio of A to B is given,
A
D
B
F
C
E
a 2. prop.
let the fame be done of the faid D to E. Now ſeeing
that D is given, (a) E is alſo given. Again, feeing that
the ratio of B to C is given, let the fame be done of
E to F. But E is given, and therefore F is alfo given.
But ſeeing that D is given, (b) the ratio of the fame D b 1. prop.
to F is given; and feeing that as A to B, fo is D to E,
and as B to C, fo is E to F; in ratio of equality, (c)
as A is to C, fo is D to F; but the ratio of D to F
is given. Therefore the ratio of A to C is alſo
given.
C 22 5-
PROP.
256
EUCLIDE'S DATA.
2.8. prop.
b 8. prop.
2 11. def.
b 6. prop.
101010
D
E
F
PROP. IX.
If two or more magnitudes A, B,
and C, are to one another in a given
ratio, and that the fame magnitudes
A, B, and C, have to other magnitudes
D, E, and F, given ratio's, although
they be not the fame, thoſe other mags
nitudes D, E, and F ſhall be alſo to
one another in given ratio's.
Demonftr. Forafmuch as the ratio of A to B is gi
ven, as alfo that of A to D, the ratio of D to B fhalt
be given: But the ratio of B to E is alfo given;
therefore the ratio of the fame D to E fhall be in like
manner given. Again, ſeeing that the ratio of B to
C is given, and alſo that of B to E, the ratio of E
to C fhall be given. But the ratio of C to F is alfo
given. Therefore (a) the ratio of E to F fhall be gi-
But it hath been demonftrated that the ratio of
D to E is alfo given; and therefore (6) the ratio of D
to F fhall be given. Therefore the magnitudes D, E,
and F are to one another in given ratio's.
ven.
PROP. X.
D B
C
If a magnitude A B, bë
greater than another mag-
nitude BC. by a given mag
nitude, and in ratio, the magnitude AC compounded of
both, shall be alfo greater than that the fame magnitude,
by a given magnitude, and in ratio; but if that compound-
ed magnitude be greater than the jame magnitude, by a
given magnitude, and in ratio; either the remainder
Shall be also greater than that fame, by a given magnitude,
and in ratio; or else the fame remainder is given with
the following, to which the other magnitudes hath a given
ratio.
Demonftr. For feeing that AB is greater than BC by
a given magnitude, and in ratio, let the given magni-
tude AD be taken away. Therefore (a the reaſon of
the remainder DB to BC is given; and by compound-
ing, (b) the ratio of DC to BC is alfo given. But the
magnitude AD is alfo given; therefore AC is greater
than the fame BC by a given magnitude, and in ratio.
Again,
EUCLIDE'S DATÀ:
257
Again, Let the mag-
hitude AC be greater
than the magnitude BC,
A
D BE
by a given magnitude, and in ratio; I fay, that the
reft AB, is either greater than the fame BC by a given
magnitude, and in ratio; or that the fame AB, with
that which followeth, to which BC hath a given ratio,
is given.
Forafmuch as the magnitude AC is greater than the
magnitude BC, by a given magnitude, and in ratio,
cut off from it the given magnitude: Now the fame
given magnitude is either lefs than the magnitude AB,
or greater: Let it in the first place be leſs, and let it
be AD. Therefore the ratio of the remainder DC to
CB is given. Wherefore by divifion, the ratio of
DB to BC is given. But the magnitude AD is alſo
given; therefore the magnitude AB is greater (c) than
the magnitude BC by a given magnitude, and in ratio.
Now let the given magnitude be greater than the mag-
nitude AB, and let AE be put equal thereto; therefore
(d) the ratio of the remainder EC to CB is given; and
by converfion, (e) the ratio of the fame BC to BE, is
alfo given. But the fame EB with BA is given, for
that the whole AE is given: Therefore there is given
AB, with that which follows BE, to which BC hath
a given ratio.
PROP. XI.
If a magnitude AB be great-
er than a magnitude BC, hy A-
a given magnitude, and in
ED B
>
C
ratio, the fame magnitude AB, fhall be also greater than
the magnitude compounded of them by a given magnitude;
and in ratio, and if the fame magnitude be greater than
the two others together by a given magnitude, and in ratio;
that fame magnitude shall be alfo greater than the rest by
a given magnitude, and in ratio.
Demonftr. For feeing that the magnitude AB is great-
er than BC by a given magnitude and in ratio; let
there be taken from it a given magnitude AD: There-
fore (a) the ratio of the reft DB to DC, is given, and
therefore (b) the ratio of DC to BD fhall be alfo given :
Let the fame be done of AD to DE, therefore the ratio
of the fame AD to DE is given: But AD is given,
therefore (c) DE is alſo given, and confequently, (d) the
R
ret
c 11. def.
dii. def.
e 5. prop:
ail. cf.
b 6.
prop
c 2. prof.
d 4. pruto
258
EUCLIDE'S DATA.
´e 16. 5.
f 18. 5.
g 16. 5.
h 11. def.
i 11. def.
k 5. prop.
1.2. prop.
m 19. 5.
n ſchol. 5.
prop.
0 11. def.
reft AE, is alfo given. But feeing that as AD : DE': :
DC: BD; by permutation, (e) as AD: DC: DE :
DB: Therefore by compounding, (f) as AC: CD : :
EB: DB; and by permutation, (g) as AC: EB:: DC:
DB. But the ratio of DC to DB is given: Therefore
alfo is AC to EB, and confequently that of EB to AC.
But it hath been demonftrated that AE is given, there-
fore (b) AB is greater than AC by a given Magnitude,
and in ratio.
But now let AB be greater than AC by a given Mag-
nitude, and in ratio: I fay, that the fame AB is alſo
greater than the reft BC by a given Magnitude, and in
ratio.
For feeing that AB is greater than AC by a given
Magnitude, and in ratio. Let the given Magnitude
AB be cut off therefrom: Therefore (i) the ratio of the
remainder EB to AC is given, and confequently alſo
ſhall be given that of AC to EB. Let the fame be done
of AD to DE, therefore the ratio of AD to DE is gi-
ven; and by converfion, (k) the ratio of AD to AE fhall
be alfo given, and confequently that of AE to AD. Now
AE is given, therefore the whole AD (/) ſhall be alſo
given; and feeing that as the whole AC is to the whole
EB, fo the part cut off AD, is to the part cut off ED,
fo alfo fhall be (m) the remainder DC to the remainder
DB. But the ratio of AC to EB is given: Therefore
alſo ſhall be given that of DC to DB. Wherefore by
divifion. (2) the ratio of BC to DB is given; and con-
fequently alſo fhall be given that of DB to BC. But it
hath been demonftrated that AD is given: Therefore (0)
AB is greater than the fame BC by a given Magnitude,
and in ratio.
L
B
C
-1-
PROP. XII.
If there are three magnitudes
·D AB, BC, and CD, and that the
firft AB, with the fecond BC, to
wit, AC, be given. And the fecond BC, with third CD,
to wit, BD, be alfo given: Either the firft AB fhall be
equal to the third CD, or the one ſhall be greater than the
other by a given Magnitude.
Demonftr. Forafmuch as each of the Magnitudes AC
and BD are given, the given Magnitudes are either equal
to one another, or unequal. Let them be firſt equal :
Therefore AC is equal to BD; take away the common
Magnitude
EUCLIDE'S DATÁ.
259
B C
A]—]
E
a
Magnitude BC, and there will remain (a) AB, equal à 3. ax. 1.
to CD. But fuppofe them to be unequal as in this fe-
cond figure, and let BD be greater than AC: Let then
BE be put equal to AC.
Now feeing that AC is gi-
ven, BE is alfo given. But
the whole BDis alfo given,
the reft ED (b) fhall be fo alfo; and forafmuch as BE is
equal to AC, taking away the common Magnitude (c)
BC, there will remain AB equal to CE: But ED is
given; Therefore CD is greater than AB by the given
Magnitude ED.
1
PROP. XIII.
C-
If there are three magnitudes AB,
CD, and E, and that the first of A
them AB, bath a given ratio to the
Second CD; but the fecond CD is C-
greater than the third E, by a gi-
ven magnitude, and in ratio; alfo
the first AB, fhall be greater than
the third E, by a given magnitude, and in ratio.
H
-1-
B
F
D
E
Demonftr. For feeing that CD is greater than E by a
given Magnitude, and in ratio; let the given Magni-
tude CF be taken therefrom: Therefore the ratio of
the rest FD to E is given. And forafmuch as the ratio
of AB to CD is given, let the fame be done of AH to
CF. Therefore the ratio of the fame AH to CF is
given. But CF is given: Therefore (a) AH is alſo given.
And ſeeing that as the whole AB is to the whole CD,
fo is the part cut off AH is to the part cut off CF, and fo
(6) alfo the reft HB is to the reft FD, the ratio of the
fame HB to FD is alfo given. But the ratio of FD to
E is alſo given: Therefore (c) the ratio of HB to F is
given. But it hath been demonftrated that AH is gi-
ven: Therefore (d) AB is greater than the faid E by a
given Magnitude, and in ratio.
b 4. prop.
c z.ax. 1.
a 2. propi
b 19. 5.
c9. prop.
d 11. def:
PROP.
260
EUCLIDE'S DATA.
a 1. prop.
b 12.5.
c 2. prop.
d 4. prop.
e 12.5.
f 11. def.
PROP. XIV.
B G
Al
-E
D
C
1.F.
If two Magnitudes AB.
and CD, have to one ano-
ther a given ratio, and
that to each of them there
be added a given Magnitude,
to wit, BE and DF; either the whole AE and CF fhall
have to one another a given ratio, or the one shall be greater
than the other by a given Magnitude, and in ratio.
Demonftr. For feeing that each of thofe Magnitudes
BE and DF, is given, (a) the ratio of the faid BE
and DF is alfo given; and if that ratio be the fame
with that of AB to CD, that of the whole AE to the
whole CF, (b) fhall be the fame; and therefore the ratio
of the faid AE to CF is given.
鄒
​Now let the ratio of BE to DF be not the fame
with that of AB to CD, and let it be as AB to CD, fo is
BG to DF. Therefore the ratio of the faid BG to
DF is given. But the Magnitude DF is given, there-
fore (c) BG is alſo given; and feeing that the whole BE
is given, (d) the reft GE fhall be alfo given. But for-
afmuch as AB is to CD, as BG is to DF, (e) fo alſo is the
whole AG to the whole CF; and therefore the ratio of
the ſaid AG to CF is given: But the Magnitude GE is
given: Therefore (f) the Magnitude AE is greater than
the Magnitude CF by a given Magnitude, and in
ratio.
PROP. XV.
E G
-B
F
Com
D
If two Magnitudes AB and
CD, have to one another a given
ratio, and that from each of
them be taken away a given
Magnitude (to wit, from the
Magnitudt AB the Magnitude AE, and from the Magni-
tude CD the Magnitude CF) the remaining Magnitudes EB
and FD, either shall have to one another, a given ratio,
or the one of them ſhall be greater than the other by a given
Magnitude, and in ratio.
Demonflr.
1
EUCLIDE'S DAT A.
261
Demonftr. For feeing that each Magnitude AE and CF
is given, the ratio of AE to CF is given; and if it be
the fame with that of AB to CD, that of the remainder
EB to the remainder FD, (a) ſhall be alſo the fame ; and
therefore the ratio of the faid EB to FD fhall be alfo
given. But if it be not the
a 19. 5.
E G
Now the ratio of A-—1———————1—————————-B
fame, let it be as AB to CD, fo
is AC to CF.
AB to CD is given, therefore
alfo that of AG to CF fhall be C-
given. But CF is given, there-
F
-D
fore (6) AG is given. But AE is alſo given, therefore (c)
the reſt EG is given; and feeing that as AB is to CD,
fo the part cut off AG is to the part cut off CF, and ſo
alfo is (d) the reft GB to the reft FD; the ratio of the
faid GB to FD is alfo given. Therefore ſeeing that EG
is given, EB is greater than FD (e) by a given Magnitude,
and in ratio.
b 2. prop.
c 4. prop.
d 4. prop.
e 11. def.
PROP. XVI.
If two Magnitudes AB and
CD, have to one another a given
ratio, and that from one of them, A
to wit, CD, there be taken away
G
P
E
a given Magnitude DE, and to C-
the other AB there be added a
-D
given Magnitude BF, the whole AF ſhall be greater than the
reft CE, by a given Magnitude, and in ratio.
Demonftr. For feeing that the ratio of AB to CD is
given, let the fame be made of BG to DE: Therefore
(a) the ratio of the faid BG to DE is given. But DE is
given, therefore (5) BG is alfo given. But BF is alſo
given, therefore (c) the whole GF is given. And feeing
that as AB is to CD, fo the part cut off BG, is to the part
cut off DE; and (d)fo alfo is the remainder AG to the
remainder CE; the ratio of the ſaid AG to CE is given,
But GF is given, therefore the Magnitude AF is greater
than the Magnitude CE by by a given Magnitude, and
in ratio.
a 2. def.
b 2. prop.
c 3. prop.
d 19. 5.
PROP.
R 3
262
EUCLIDE'S DATA.
a 8. prop.
b 14. prop.
2
propo
b 12. 5.
E-
F
A
-1-
C-
-1-
G
RPO P. XVII.
If there are three Magnitudes
AB, E, and CD, and that the
B firft AB be greater than the fe
cond E by a given Magnitude,
and in ratio: And the third
CD be alfo greater than the
fame fecond E, by a given Mag-
the first AB fhall have to the third
CD, either a given ratio, or else the one shall be greater
than the other by a given Magnitude, and in ratio.
nitude, and in ratio;
D
Demonftr. For feeing that AB is greater than E by a
given Magnitude, and in ratio, let the Magnitude AF
be taken away: Therefore the ratio of the remainder
FB to E is given. Again, feeing that CD is greater than
the faid E by a given Magnitude, and in ratio, let the
given Magnitude CG be cut off therefrom; and the ra-
tio of the remainder GD to E fhall be given: Therefore
(a) the ratio of FB to GD fhall be alfo given. But to
the faid FB and GD are added the given Magnitudes
AF and CG: Therefore the whole AB and CD (6) fhall
either have to one another a given ratio, or the one
fhall be greater than the other by a given Magnitude,
and in ratio.
B
A
I
C
1
F
E
PROP. XVIII.
G
-H
If there are three Magnitudes
AB, CD, and EF, and that the
one of them, to wit, CD, be
1-D greater than either of the other
AB or EF, by a given Magni-
tude, and in ratio; either the
two others AB and EF, fhall have to one another a given
ratio, or the one shall be greater than the other by a given
Magnitude, and in ratio.
-K
Demonftr. Forafmuch as the Magnitude CD is greater
than the Magnitude AB by a given Magnitude, and in
ratio, let the given Magnitude LG be taken therefrom:
Therefore the ratio of the remainder CG to AB is
given. Let the fame be made of GD to BH, therefore
the ratio of the faid DG to BH is given. But DG is
given, therefore (a) BH is alfo given. And feeing that as
CG is to AB, fo is GD to BH, (b) fo alfo is the whole
CD to the whole AH, the ratio of the faid CD to AH
fhall be alfo given.
Again,
EUCLIDE'S DATA.
263
Again, feeing that the fame CD is greater than EF by
a given Magnitude, and in ratio; let the Magnitude
c 12.5.
DI be cut off therefrom: Therefore the ratio of the re-
mainder CI to EF is given: Let the fame be made of DI
to FK. Therefore the ratio of the faid DI to FK ſhall
be alſo given. But DI is given, therefore FK is alſo gi-
ven. And ſeeing that as CI is to EF, fo is ID to FK ;
fo alfo is the whole (c) CD to the whole EK.; the ratio
of the faid CD to FK fhall be given. But the ratio
of the fame CD to AH is alfo given: Therefore (d) the
ratio of the faid AH to EK fhall be given. And feeing
that from the ſaid AH and EK, the given Magnitudes
BH and FK are cut off, the Magnitudes AB and EF (e) e 15. prop.
are either in a given ratio to one another, or the one is
greater than the other by a given Magnitude, and in ratio.
PRO P. XIX.
If there are three Magnitudes
AB, CD, and E, and that the A-
GH
B
firft AB be greater than the fe-
F
cond CD, by a given Magnitude, C-
1
-D
and in _ratio; and that the fe-
E
d 8. prop.
1
cond CD be greater than the
third E, by a given Magnitude,
and in ratio; alfo the firft Magnitude AB fhall be greater
than the third E, by a given Magnitude, and in ratio
Demonftr. For feeing that CD is greater than E by a
given Magnitude, and in ratio; let the given Magni-
tude, CF be taken therefrom: Therefore the ratio of
the remainder FD to E is given. Again, feeing that AB
is greater than the fame CD by a given Magnitude, and
in ratio: Let the Magnitude AG be taken therefrom:
Therefore the ratio of the remainder GB to CD is
given: Let the fame be made of GH to CF. Therefore
the ratio of the faid GH to CF is given. But CF is
given: Therefore alfo GH is given, and then AG is alſo
given, the whole_(a) AH fhall
be alfo given. But as GB is
to CD, fo is GH to CF, and A-
fo alfo (b) the remainder HB to
the remainder FD: Therefore C
the ratio of the ſaid HB to
FD is given. But the ratio of
23. prop.
G H
F
b 19.5.
1
D
E
the fame FD to E is alfo given :
Therefore the ratio of HB to E is in ixA
R 4
264
EUCLID E's DATA.
1
C11. def.
d 13. prop.
2. prop.
a 1. prof.
$ 19.5.
given, and fo is alío the Magnitude AE: Wherefore
the Magnitude AB (c) is greater than E by a given Mag-
nitude, and in ratio.
A-
E F
C
D
OTHERWISE.
1 B
Conftruction. Let there be
three Magnitudes AB, C, and
D, and let AB be greater than
C by a given Magnitude, and
in ratio; but let C be alfo
greater than D, by a given
Magnitude, and in ratio:
I fay, that AB is greater than D by a given Magnitude,
and in ratio.
Demonftr. Forafmuch as AB is greater than C by a
given Magnitude, and in ratio, let the given Magni-
tude AE be cut off therefrom: Therefore the ratio of
the remainder EB to C is given. But the Magnitude C
is greater than the Magnitude D by a given Magnitude,
and in ratio; therefore (d) EB is greater than D by a
given Magnitude, and in ratio: Wherefore let the gi-
ven Magnitude EF be cut off therefrom; and the ratio
of the remainder FB to D fhall be given. But AF is
(e) given. Therefore AB is greater than D by a given
Magnitude, and in ratio,
PROP. XX.
E
G
A-
-1
F
C—
-D
If there are two given
B Magnitudes, AB and CD,
and that from them there
are taken Magnitudes AE
and CF, having to one ano-
ther a given ratio; either the remaining Magnitudes EB
and FD, shall have to one another given ratio's ; or elſe the
one ſhall be greater than the other by a given Magnitude,
and in ratio.
Demonftr. For feeing that both the Magnitudes AB and
CD, are given, the ratio of the faid AB to CD is (a) alſo
given; and if it be the fame as of AE to CF, that of the
remainder ´EB to the remainder FD fhall be (b) alſo the
fame; and therefore the ratio of the faid EB to FD fhall be
alfo given. But if it be not the fame, let it be ſo as that
AE be to CF, as AG to CD. New the ratio of the
faid AE to CF is given: Therefore the ratio of the faid
·
AG
EUCLIDE'S DAT A.
AG to CD is given. But CD is given, therefore (c) AG
is alfo given. But the whole AB is likewife given,
therefore (d) the remainder BG is given. And feeing that
as AE is to CF, fo is AG to CD, and alſo the remainder
EG to the remainder FD, the ratio of the faid EG to
FD is given. But GB is alfo given: Therefore the Mag-
nitude EB is greater (e) than the Magnitude BD by a gi-
ven Magnitude, and in ratio.
PROP. XXI.
If there are two Magnitudes gi-
wen, AB and CD, and to them are A-
added other Magnitudes BE and
DF, having to one another a given
ratio; either the whole AE and
C
G B
D
IF
CF shall have to one another a given ratio, or else the one
Shall be greater than the other by a given Magnitude, and
in ratio.
265
c 3 prop.
d 4. prosi
c 11. def.
a 1. prop.
Demonftr. For feeing that both the Magnitudes AB
and CD are given: their ratio (a) is alfo given, and if
it be the fame ratio as of BE to DF, the ratio of the
whole AE to the whole CF fhall be alſo given; for it
fhall be (b) the fame. But if it be not the fame, let it b 12.5.
be as BE is to DF, fo BG to CD: Therefore the ratio
of the faid BG to CD is given. But CD is given;
therefore (c) alfo BG fhall be given. But the whole AB
is given; therefore alfo the (d) remainder AG fhall be gi-
ven. And ſeeing that as BE is to DF, ſo is BG to CD,
and alfo (e) the whole GE to the whole CF, the ratio of
the ſaid GE to CF ſhall be likewife given. But AG is gi-
ven; therefore the Magnitude AE is grearer than the
Magnitude CF by a given Magnitude, and in ratio.
PROP. XXII.
If two Magnitudes AB and A-
BC, have to fome other Magni-
tude D, a given ratio, alſo
their compound Magnitude AC,
B
-1.
C
D
fhall have to the fame Magnitude D, a given ratio.
Demonflr. For feeing that each Magnitude AB and
BC hath a given ratio to D, the ratio (a) of AB to
BC is given; and by compounding, (b) the ratio of AC
to BC is given. But that of BC to D is alfo given,
therefore (c) the ratio of the faid AC to D fhall be like-
wife given,
PROP,
c 2. prop.
d 4. prop.
e 12. 5.
:
a & prop.
b 6. prope
c 8. prope
266
EUCLIDE'S DATA.
a 19. 5.
b 8. prop.
€ 5. prop.
d 8. prop.
e 5. prop.
£ 8. prop.
£ 5. prop.
h 8. prop.
!
E
PROP. XXIII.
-B
F G
C————————————▬▬▬▬▬▬▬-D
If the whole AB be to the
whole CD in a given ratio, ana
that the parts AE and EB be to
the parts CF and FD in given.
ratio's, altho' they be not the
fame, the whole (to wit, AB, AE, and BE,) ſhall be to the
whole (to wit, CD, CF, and FD,) in given ratio's.
Demonftr. For feeing that AE is to CF in a given ra-
tio, let the fame be made of AB to CG; therefore the
ratio of the ſaid AB to CG is given; and confequently
alſo that (a) of the reft EB to the reft FG. But the ratio
of FD to the fame EB is alſo given: Therefore the ratio
of FD to FG (6) is likewife given, and therefore (c) that
of FD to the remainder GD is alfo given. But the ratio
of AB to each of the Magnitudes CD and CG is given:
Therefore (a) alſo the ratio of CD to CG is given, and
again (e) that of CD to the remainder GD. But the ra-
tio of FD to DG is given, therefore alſo (ƒ) that of the
fame CD to FD, and confequently that of (g) CD to the
remainder FC; and therefore
alſo the ratio of CF to FD fhall
B be given. But the ratio of EB
to FD is propoſed to be given;
therefore the ratio of CF to EB
fhall be given. Again, for that
the ratio of AB to CD is given; and alfo that of CD
to each of thofe FC and FD, the ratio of the fame AB
to each of the faid FC and (b) FD, ſhall be likewiſe given.
But the ratio of the faid FD to EB is given: Therefore
the ratio of AB to BE fhall be alſo given, and confe-
quently AB to the remainder (i) AE. Wherefore by divi-
fion (A) the ratio of AE to EB fhall be likewife given.
But the ratio of EB to FD is given. Therefore alſo
that of AE to FD. In like manner, ſeeing that the ra-
tio of CD to AB is given; and that of AB to each of
his parts AE and EB; alfo the ratio of the faid CD
to each of the faid AE and EB, (/) fhall be given: Where-
fore each of the Magnitudes AB, CD, AF, EB, CF, and
FD, is to each of the others in a given ratio.
i 5. prop.
k ſch. 5. pr.
18. prop.
E
A-1-
F G
-D
PROP.
EUCLIDE'S DATA.
267
PROP. XXIV.
If of three right-lines A, B, and
C, proportional A to B, as B to C, the
firft A hath to the third C a given
ratio, it will also have to the fecond B
a given ratio.
Demonftr. For, let there be expo-
fed another right-line D, and feeing
A
D
B
E
C
F
that the ratio of A to C is given; let the fame be made
of D to F; therefore the ratio of D to F is given. But
Dis given, therefore F is alfo given; betwixt the two
right-lines D and F, let there be taken (a) a mean propor-
tional E. Therefore the rectangle made under D and F
is equal (b) to the fquare of E. But the fame rectangle of
D and F is (c) given: (for all the angles of that rectangle
are given, being right-angles, and the ratio's that the
fides have to one another are alfo given ;) therefore the
fquare of E is given, and confequently the fame right-
line E is alſo given (for one equal thereto may be found,
(4) feeing that the rectangle of D and F is given.) But D
is given, therefore (e) the ratio of D to E is given, and
as A is to C, fo D is to F. But as A is to C, (f) fo is
the fquare of A to the rectangle of A and C, and alfo
as D is to F, fo is the fquare of D to the rectangle of
D and F. Therefore as the fquare of A is to the rectan-
gle of A and C, fo is the fquare of D to the rectangle of
D and F. But the rectangle of A and C is equal to the
fquare of B, (feeing that A, B, and C, are proportional)
and that of D and F to the ſquare of E, therefore as the
fquare of A is to the fquare of B, fo is the fquare of D
to the fquare of E Wherefore (g) as A is to B, fo is D
to E. But the ratio of D to E is given, therefore (b)
alfo the ratio of A to B is given,
PRO P. XXV. Plate VII. Fig. 29.
If two lines AB and CD, given by pofition do interfect,
the point E in which they cut one another, is given by
pofition,
Demonftr. For if it change its place the one or the
other of the lines AB and CD, would change its
pofition: But fo it is that by Suppofition it changeth
not: : Therefore (a) the point E is given by poſition.
PROP.
2 13.76
b 17. 6.
© 3. def.
d 14. 2.
e 1. prop.
f1. 6.
g 12.6.
h 2. def.
a 4, def.
268
EUCLIDE'S DATA.
a 6. def.
b 25. prop.
AB
right-line AB is
PRO P. XXVI.
If the extremeties A and B, of a right-
line AB, are given in pofition, that fame
given in pofition and in magnitude.
Demonftr. For if the point A remaining in its place,
the pofition, or the magnitude of the right-line AB fhall
change, the point B will fall elſewhere. But ſo it is,
that by Suppofition it doth not fall elfewhere. Therefore
the right-line AB is given in pofition, and in magni-
tude.
A- -B
PRO P. XXVII.
If one of the extremes A of a right-line
AB, given in pofition and magnitude, be
given, the other extremity, В shall be also given.
B
Demonftr. For if the point A remaining in its place,
the point B fhall change and fall in fome other place,
either the pofition of the right-line AB, or its magnitude
would change: But fo it is that according to the Suppo-
fition, neither the one nor the other doth change.
Therefore the point B is given.
OTHERWISE.
Confir. On the center A, (Fig. 30.) with the diſtance
AB, defcribe the circumference BC.
Demonftr. Therefore (a) that circumference BC is
given by pofition. But the right-line AB is alfo
given by pofition; therefore the point (6) B is given,
PRO P. XXVIII. Plate VII. Fig. 31.
If through the given point A, there be drawn a right-
line DAE, against another right-line BC, given in po-
fition, the right-line DAE fo drawn, is given in pofition.
Demonftr. For if it be not given, the point A re-
maining in its place, the pofition of the right-line DAE
may change Let it then change if it be poffible, and
fall elsewhere, remaining parallel to BC, and let it be
the line FAG: Therefore BC is parallel to the faid line
FAG,
EUCLIDE'S DATA.
269
FAG. But (a) the fame BC is alfo parallel to DAE:
Therefore (b) DAE is parallel to the faid line FAG,
which is abfurd; feeing they join together, and meet in
A: Therefore the pofition of the right-line DAE falls
not elsewhere. Wherefore the faid line DAE is gi-
ven in pofition.
PRO P. XXIX. Plate VII. Fig. 32.
If to a right-line AB, given in pofition, and to a
point C given therein, there be drawn a right-line CD,
which shall make a given angle ACD, the line drawn
CD is given in pofition.
Demonftr. For if it be not given in pofition, the point
Cremaining in its place, the pofition of the line CD
obferving the Magnitude of the angle ACD, will fall
elſewhere. Let it fall elſewhere then if it be poffible,
and let it be CE. Therefore the angle ACD is equal to
the angle ACE, the greater to the leffer, which is ab-
furd. Therefore the pofition of the right-line CD, fhall
not fall elſewhere; and therefore the ſaid line CD is gi-
ven in poſition.
,、་
PRO P. XXX. Fig. 33.
a
If from a given point A, be drawn to a right-line
BC, given in pofition, right-line A D, making a
given angle ADB, that line drawn AD is given in
pofition.
Demonftr. For if it be not given, the point A re-
maining in its place, the pofition of the right-line AD
changing, the magnitude of the angle ADB, will change.
Let it change then, and let it be the right-line AE: There-
fore the angle ADB is equal to the angle AEB, the
greater (a) to the leffer, which is abfurd. Therefore
the pofition of the right-line AD doth not change;
and therefore the faid line AD is given in pofition.
OTHERWISE.
Confir, Through the point A let there be drawn
the line. EAF, parallel to the right-line BC.
Demonfr. Then feeing that through the given point A,
and against the right-line BC, given in pofition, there is
drawn
a 13. def.
३१.
b
a 16. 1.
J
1
870
EUCLIDE'S DATA.
b 29. 1.
€
drawn the right-line EF, (Fig. 34.) thofe lines EF and
BC are parallels. But on the fame lines doth alſo fall
the right-line AD. Therefore (b) the angle FAD is equal
to the given angle ADB; and therefore it is alfo given.
Wherefore to the right-line EF given in pofition, and to
the given point A therein, there is drawn the right-line
29. prop. AD, making the given angle FAD. Therefore (c) the
faid line AD is given in pofition.
d29. 1.
€
OTHERWISE.
Conftr, In the line BCE, (Fig. 35.) let there be taken
the given point C, and through the fame let there be
drawn the line CF, parallel to the ſaid DA.
Demonftr. Forafmuch as AD and CF are parallels,
and that on them there falls the right-line BCE, the
angle FCB is equal (d) to the given angle ADB; and
therefore it is alſo given. And feeing that the right-line
BC is given in pofition, and that to a given point C
therein, there is drawn the right-line FC, making the
29. prop. given angle FCB, that fame line FC (e) is given in
pofition. But through the given point A, oppofite to
the line FC given in pofition, there is drawn the line
AD. Therefore the faid line (f) AD is given in
pofition.
f 28. prop.
B
26.
prop.
OTHER WISE.
Conftr. In the right-line BC (Fig. 36.) aflume fome
point at F, and draw AF.
Demonftr. Forafmuch as each point A and F is given,
the right-line AF is given (g) in pofition. But the line
BC is alfo given in pofition. Therefore the angle
AFD is given. But by fuppofition, the angle ADF
is given: Therefore DAF (which is the refidue (b) of
two right-angles) is given; and feeing that to the right-
line AF given in pofition, and to the given point therein
A there is drawn the right-line DA, making the given
i 29. prop. angle DAF, (i) that fame line DA is given in poition.
h32. 1.
Scholium.
* EUCLIDE fuppofeth here that two right-lines being
given in poſition, and inclining to one another, make a
given angle, which ſome demonſtrate after this manner.
Demon r
EUCLIDI's DATA.
371
Demonftr. Forafmúch as the two right-lines given in
pofition, incline to one another, the inclination of
thofe lines is given. But the angle is the inclination
of the lines: Therefore the angle which makes the
right-lines given in pofition, and inclining to one ano-
ther, is given.
Another thus demonftrates it.
Conftr. Let there be two right-lines inclining to one
another, as AB and CB, (Fig. 37, 38.) given in pofition,
and in the line AB let there be taken a given point A,
and in BC alſo ſome point, as C; and let the right-line
AC be drawn.
Demonftr. Seeing that as well the point B, as each of
the points A and C is given, (4) the three right-lines k 26. propi
AB, BC, and AC, are given in Magnitude. Where-
fore of three direct lines equal unto them, a triangle
may be conftituted: Let there then be made the tri-
angle FDE, having the fide FD equal to the fide AB,
the fide FE equal to the fide AC, and the baſe DE
equal to the baſe BC.
Seeing then the angles compriſed of equal right-lines
are equal, we have found the angle FDE equal to the
angle ABC; and therefore the fame (1) angle ABC is
given.
PROP. XXXI. Plate VII. Fig. 39.
If from a given point A there be drawn to
right-line given in pofition BC, a right-line AD, given
in magnitude, that line AD fhall be also given in pofition.
Conftr. From the center A, with the diſtance AD, let
the circle DEF be defcribed.
Demonftr. Forafmuch as the center A is given in pofi-
tion, and the femidiameter AD in magnitude, the circle
'DEF (a) is given in poſition. But the right-line BC is
alfo given in pofition: 'Therefore the point of interfection
D() is given, and feeing that the point A is alfo given;
() the right-line AD is given in pofition.
PRO P. XXXII. Fig. 40.
If unto parallel right-lines AB and CD, given in pofition,
there be drawn a right-line EF, making the given angles
BEF and EFD, the line drawn EF shall be given in
magnitude.
Confir.
1 1. def.
a 6. def.
b25. prop.
c 26. prop.
272
EUCLIDE'S DATA.
à 29. 1.
Conftr. For let there be taken in the line CD a given
point G, and from that point let be drawn GH pa
rallel to FE.
Demonftr. Forafmuch as the lines EF and HG are
parallels, and that on them falls the line CD; (a) the
angle EFD is equal to the angle FGH. But the angle
EFD is given, therefore the angle FGH is alfo given.
And forafmuch as to the right-line CD given in pofition,
and to the point G given in the fame, there is drawn the
b 29. prop. right-line GH, making the given angle FGH, (b) the faid
line GH is given in pofition. But AB is alfo given in
pofition, therefore (c) the point H is given. But the point
G is alfo given: Therefore (d) the line GH is given in
Magnitude, and is (e) equal to EF. Wherefore (ƒ) the
faid line is given in Magnitude.
c 25. prop.
d 26. prop.
e 34. I.
f. 1. def.
a 34. 1.
b 6. def.
c 25. prop.
d 26. prop.
e-ſch..30.
prop.
f 29. 1.
g 29. 1.
PRO P. XXXIII. Plate VII. Fig 41.
If unto parallel right-lines AB and CD, given in pofition,
there be drawn a right-line EF given in magnitude, that
line EF ſhall make the given angles BEF and DFE.
Conftr. For let there be taken in the right-line AB
the point G, and through that point let there be drawn
the line GH parallel to EF.
Demonftr. Therefore EF is equal to the faid (a) GH.
But EF is given in Magnitude, therefore GH is alſo
given in Magnitude. But the point G is given, and
therefore if on that point, with the diftance GH, there
be deſcribed a circle, (b) that circle ſhall be given in po-
fition: Let it be then defcribed, and let it be HKL, the
faid circle HKL is therefore given in pofition. But the
line CD which cuts the circumference KHI, in H,
is alfo given in pofition. Therefore the faid point of
interfection H (c) is given. But the point G is given:
Therefore d) the right-line GH is given in pofition. But
the right-line CD is alfo given in pofition: Therefore
But to that angle (f) the
(e) the angle GHF is given.
angle EFD is equal; Therefore the angle EFD is given;
and therefore alſo the angle BEF; for that is only the
refidue of the ſum of two (g) right-angles.
OTHER WISE.
Conftr. Let there be taken in the right-line CD, (Fig.
42.) the point G, and let GD be put equal to EF, then
from the center G, with the diftance GD, let there be
defcribed the circle HDB, and draw GB.
Demonflr.
ま
​2.73
EUCLIDE'S DATA.
"
Demonftr. Forafmuch as the center G is given in pofi-
tion, and the femidiameter GD in magnitude, the circle
BDH (b) is given in poſition. But the line AB is alſo
given in pofition: Therefore (i) the point B is given.
But the point G is alfo given, therefore (k) the right-line
GB is given in pofition. But the right-line CD is alfo
given in pofition: Therefore (1) the angle BGD is given.
Wherefore if EF be parallel to BG, the angle EFD (m)
fhall be given, and confequently alfo the other angle
BEF. But the right-lines BG and EF being not paral-
lels, let them meet in the point H. Forafmuch as EB
is parallel to FG, and EF is equal to GD, that is to fay,
to BG; alfo FH (2) fhall be equal to GH (for EH and
BH being cut proportionally (o) by the parallel FG, as
EF is to FH, fo is BG to GH, and by permutation, as
EF is to BG, fo is FH to GH :) Therefore (p) the angle
HFG is equal to the angle HGF, but the faid angle
HGF is given (for it is equal (q) to the given angle
BGD:) Therefore the angle HFG is alfo given.
But to that angle the angle BEF is equal; and therefore
is given, as alſo the remaining angle EFG.
PROP. XXXIV. Plate VII. Fig. 43.
If from a given point E, there be drawn unto parallel
right-lines AB and CD, given in pofition, a right-line EFG,
that right-line EFG fhall be divided in a given ratio (to
wit) as EF to FG.
Conftr. For from the point E let there be drawn the
line EH, perpendicular to the line CD.
Demonftr. Forafmuch as from the given point E there
is drawn to the line CD the right-line EH, making the
given angle EHG, (a) the faid line EH is given in pofi-
tion, but both the one and the other lines AB and CD
is alfo given in pofition. Therefore (b) the points of in-
terfection K and H, are given. But the point E is alſo
given: Therefore (c) each line EK and KH is given.
Wherefore (d) the ratio of the ſaid EK to KH is given.
But as EK is to KH, fo is EF to FG; (for in the tri-
angle GEH the line KF being parallel to HG, the fides
EH and EG are cut proportionally :) Therefore the ratio
of the faid EF to FG is given.
$
OTHERWISE
h 6. def.
i 25. prop.
k 26. prop.
1 Sch. 30.
prop.
m 29. I.
n 14. 5.
0 2.6.
P 5. 1.
915. 1.
a 30. prop.
b25. prop.
c 26. proj:
d 1. prop.
274
EUCLIDE'S DATA.
a 30. prop.
b 25. prop.
c 26. prop.
d 1. prop.
€ 4. 6.
a 30. prop.
b 25. prop.
c 26. prop.
d z. 6.
e 6. prop.
£ 2. prop.
OTHERWISE. Plate VII. Fig. 44.
Conftr. To the parallel right-lines given in pofition,
AB and CD, let there be drawn from the point E the
right-line FEG: I fay, that the ratio of GE to EF is given.
Demonfr. For from the point E let there be drawn
to CD the perpendicular EH, and produced to the point
K; feeing therefore that from the point E to the right-
line CD, given in pofition, there is drawn the line EH,
making the given angle EHG, (a) the faid line EH is
given in poſition. But each line AB and CD is alfo
given in pofition: Therefore (b) each point of interſection
H and K is given. But the point E is alfo given, there-
fore (c) each of the lines EH and IK is given in Magni-
tude; and therefore (d) the ratio of the faid EH to EK
is given. But (e) as EH is to FK fo is EG to EF (for
the oppofite angles at the point E being equal, and
the lines AB and CD parallels, the triangles EHG and
EKF are equiangled; and therefore as EH is to EG,
fo is EK to EF; and by permutation as EH to EK, fo
is FG to EF.) Therefore the ratio of the faid lines EG
to EF is given.
PROP. XXXV. Fig. 45.
If from a given point A, to a right-line BC, given in
pofition, there be drawn a right-line AD, which let be di-
vided in E, in a given ratio (to wit) as AE to ED, and
that by the point of fection E there be drawn a right-line
FEG, oppofite to the right-line BC, given in pofition, the
line FG fo drawn fhall be given in pofition.
Conftr. For from the point A, let there be drawn the
line AH, perpendicular to the line BC.
Demonftr. For feeing that from the given point A
there is drawn to BC given in pofition, the right-line
AH making the given angle AHD, (a) the faid line AH
is given in pofition. But BC is alſo given in pofition :
Therefore (b) the point H is given. But the point A is
alfo given: Therefore (c) the line AH is given in mag-
nitude and in pofition. And feeing that (d) as AE is
to ED, fo is AK to KH, and that the ratio of AE to
ED is given, alfo the ratio of AK to KH is given;
and by compounding, (e) the ratio of AH to AK is given.
But AH is given in Magnitude: Therefore (ƒ) alſo
AK
EUCLIDE'S DATA.
275
AK is given in Magnitude. But AK is alfo given in po-
fition, and the point A is given: Therefore (g) the point g 27. prop.
K is alfo given, and feeing that by the faid given point K
there is drawn the line FG, oppofite to the right-line BC
given in pofition; the faid line FG (b) is given in pofition. h 28. prop.
PRO P. XXXVI. Plate VII. Fig. 46.
If from a given point A, there be drawn to a right-
line BC given in poſition, a right-line AD, and to it be
added a right-line AE, having to the fame AD a given
ratio, and that through the extremity E of the added line
AB, there be drawn a right-line FEK, oppofite to the
line BC, given in pofition, that fame line FEK fall be
given in pofition.
Conftr. For from the point A let there be drawn to
the line BC, the perpendicular AL, and let it be
prolonged to the point G.
Demonftr. Forafmuch as from the given point A, there
is drawn to the right-line BC, given in pofition, the
right-line GL,which makes the given angle GLD, (a) that
line GL is given in pofition. But BC is alfo given in
pofition, therefore (b) the point L is given; and feeing
that the point A is alfo given, the line (c) AL is given.
But forafmuch as the ratio of AE to AD, is given, and
that (d) as the faid AE is to AD, fo is AG to AL ;
(becauſe the triangles ALD and AGE are equiangled)
the ratio of AG to AL is alfo given. But AL is given
in Magnitude: Therefore (e) AG is given in Magnitude.
But it is alſo given in pofition, and the point A is given:
Therefore (ƒ) the point G is alſo given. And feeing that
by the fame given point G there is drawn the line FK,
oppofite to the right-line BC, given in pofition, (g) the
faid line FK is given in poſition.
XXXVII. Fig. 47.
PRO P. XXXVII.
If unto parallel right-lines AB and CD, given in po-
fition, there be drawn a right-line EF, divided in the
point G, in a given ratio, (to wit,) of EG to GF: and
if through the point of fection G, there be drawn op-
pofite to the right-lines AB or CD, given in pofition,
a right-line HG K, that line shall be given in pofition.
S 2
Conftr.
a 30. prop.
b 26. prop.
c 26. prop.
d 4. 6.
e 2.
prop.
£ 27. prop.
g 28. prop,
276
EUCLIDE'S DATA.
2 30. prop.
b 25. prop.
c 26. prop.
d 6. prop.
e 2. prop.
₤27. prop.
8 7.5.
A 17.5.
iz. 6.
k 2. 6.
1 18: 5.
m 22. 5.
→
Conftr. For let there be taken in the line AB the gi-
ven point L, and from that point let there be drawn
the line LN, perpendicular to CD.
Demonftr. Seeing that from the given point L, there is
drawn to the right-line CD, the line LN, making the
given angle LND, the faid LN (a) is given in pofition.
But CD is alfo given in pofition: Therefore the point
N (b) is given. But the point L is alfo given: Therefore
(c) the line LN is given; and feeing that the ratio of FG
to GE is given, and that* as FG is to GE, fo is NM to
to ML, the ratio of the ſaid NM to ML is given; and
by compounding, (d) the ratio of LN to LM is alfo
given. But LN is given in Magnitude; therefore ML
is (e) given in Magnitude. But it is alſo given in poſi-
tion, and the point L is given; Therefore the point M
(f) is alfo given. And confidering that through the faid
point M there is drawn the right-line KH, oppoſite to
the right-line CD, given in pofition, the faid line KH
is alfo given in pofition.
Scholium.
* EUCLIDE ſuppoſeth here, that as FG is to GE, fo NM
is to ML; but by another it is thus demonſtrated.
The lines EF and LN are parallels or not parallels: Let
them in the first place be parallels, and forafmuch as by
Conftruction the lines EL, FN, EF, and LN, are parallels,
EN ſhall be a parallelogram, and therefore the fide EF is
equal to the fide LN. Again, feeing that MG is parallel to
NF, and GF to MN, GN fhall be alfo a parallelogram; and
therefore the fide GF is equal to the fide MN. Wherefore the
equal fides EF and LN, ſhall have to the equal fides F G
and MN, (g) one and the fame ratio. Therefore as EF is
to FG, fo is LN to MN; and by dividing, (h) as GE to
GF, fo is LM to MN.
(Fig.
Now fuppofe that the lines EF and LN (Fig. 48.) are
not parallels, but that they meet in the point Ō. Foraf-
much as in the triangle OFN there is drawn HK, pa-
rallel to FN one of the fides; (i) the fides OF and ÓN
are divided proportionally; and therefore as FG is to GO,
fo is NM to MO. Again, feeing that in the triangle
OGM there is drawn EL, parallel to the fide GM, the
fides OG and OM are divided proportionally: Where-
fore (k) as OE is to EG, fo is OL to LM, and by
compounding, (1) as OG is to EG, fo is OM to LM ;
but it hath been demonftrated that as FG is to GO, ſo
is NM to MO; therefore in ratio of equality, (m) as FG
is to GE, ſo is NM to ML.
PROP.
EUCLIDE'S DATA.
277
PRO P. XXXVIII. Plate VII. Fig 49.
If unto parallel right-lines AB and CD, there be drawn
a right-line EF, and that to it there be added fome other
right-line EG, which hath a given ratio to the fame EF;
and if through the extremity G of the added line EG, there be
drawn a right-line HK, against the parallels given in pofi-
tion AB and CD, the line drawn HK ſhall be alſo given
in pofition.
Conftr. For let there be taken in the line AB, the
given point N, and from thence let there be drawn to
CD the perpendicular NM, and let it be prolonged to the
point L.
Demonftr. Forafmuch as from the given point N there
is drawn to the right-line CD, given in pofition, the
right-line NM making a given angle NMF, the faid
angle NMF (a) is given in pofition. But the line CD
is alfo given in pofition: Therefore (b) the point M is
given. But the point N is alfo given: Therefore (c) the
line NM is given, and therefore the ratio of EG to EF
is given; and becauſe (d) as EG to EF, fo is LN to MN,
the ratio of LN to NM is alfo given: But NM is given,
therefore LN is (e) alfo given. But the point N is
given: Therefore (f) the point L is alfo given. Seeing
Seeing
then that by the given point L there is drawn the right-
line HK, oppofite to the line AB given in pofition, (g)
the faid line AK is alfo given in pofition.
PRO P. XXXIX. Fig. 50.
If all the fides of a triangle ABC are given in magnitude,
the triangle is given in kind.
Conftr. For, let there be expofed the right-line DG
given in pofition, ending in the point D; but being in-
finite towards the other part G, and therein let be taken
DE, equal to AB.
Demonftr. Now feeing the faid AB is given in magni-
tude, DE is ſo alfo ; but the fame DE is alſo given in
pofition, and the point D is given: Therefore (a) the
point E is given.
Again, Let EF be put equal to BC; and ſeeing that
BC is given in magnitude, EF fhall be fo alfo. But the
faid EF is in like manner given in pofition, and the
point E is given: Therefore (b) the point F is given.
Furthermore, Let FG be taken equal to AC. Now
forafmuch as the faid AC is given in magnitude, FG is
a 30. prop.
b 25. prop.
c 26. prop.
d ſch. 37•
prop.
e 2. prop.
f 27. prop.
g 28. prop.
a 27. prop.
b 27. prop
S 3
fo
278
EUCLIDE'S DAT A.
c 6. def.
d 6. def.
e 25. prop.
£ 26. prop.
g 1. prop.
h ſch. 30.
prop.
i 15. def.
fo alfo. But FG is alfo given in pofition, and the point
F is given Therefore the point G is alfo given. Now
from the center E, with the diſtance ED, let there be de-
ſcribed the circle DHK, (c) and that circle ſhall be given
in pofition. Again, on the center F, and diftance FG,
let there be defcribed the circle GLK. Therefore (d) the
faid circle GLK is given in pofition; and therefore (e) the
point of Interfection K is given. But each of the points
E and F is given: Therefore each line (ƒ) EK, EF, and
FK, is given in pofition and magnitude. Therefore the
triangle FK is given in kind; but it is equal and
alike to the triangle ABC; and therefore the triangle
ABC is alfo given in kind.
Scholium.
manner
* EUCLIDE ſuppoſeth here, that a triangle whofe fides
are given in magnitude and pofition, is given in kind;
but the antient Interpreters demonftrated it in a
thus. Forafmuch as the right-lines KE and EF are gi-
ven, (g) the ratio which they have to one another is alfo
given. Also the right-lines EF and FK being given,
their ratio is also given; and in like manner, the ratio
of the faid EK and FK is given. Again, ſeeing that
the fame lines KE and EF are given in pofition, (h)
the angle KEF is given in magnitude: Moreover, the
right-lines EF and FK being given in pofition, the an-
gle EFK is given in magnitude, as is also the refidue
EKF, and fo in the triangle EKF are all the angles
given, and alfo the ratio's of the fides: Therefore (i) the
faid triangles EKF is given in kind.
PRO P. XL. Plate VII. Fig. 37, 38.
If the angles of a triangle ABC, are given in mag-
nitude, the triangle is given in kind.
Conftr. Let there be expofed the right-line DE, gi-
ven in pofition and in magnitude; and let there be
conſtitued at the point D the angle EDF, equal to
the angle CBA, and at the point E the angle DEF,
equal to the angle BCA; therefore the third angle
BAC is equal to the third angle BFE.
Demonftr.
EUCLIDE'S DATA.
279
Demonftr. For each of the angles conftituted in the
points A, B, and C, is given: Therefore each of thoſe
which are pofited in the points D, F, and E, is alſo given;
and feeing that to the right-line DE given in pofition,
and to the point D given therein, there is drawn the
right-line DF, which makes the given angle EDF, (a)
the line DF is given in poſition; and for the fame reaſon
the line EF is given in pofition: Therefore (b) the point
F is given in pofition. But each of the points D and
E is given: Therefore (c) each of the lines DF, DE, and
FF is given in magnitude. Wherefore the triangle
DFE is given in kind; and is alike to the triangle ABC:
Therefore the triangle ABC is given in kind.
PRO P. XLI. Plate VII. Fig. 37, 38.
If a triangle ABC, hath one angle BAC given, and that
the two fides BA and AC, which conftitute it, have to
one another a given ratio, the triangle is given in kind.
Confir. For, let there be expofed the right-line DF gi-
ven in magnitude and pofition. And thereon, and at the
given point F, let there be conftitued the angle DFE
equal to the angle BAC.
Demonftr. Now the angle BAC is given: Therefore
alfo the angle DFE is given, and feeing that to the
right-line DF given in pofition, and from the given
point F therein is drawn a right-line FE, making the
given angle DFE, (a) the ſaid line FE is given in pofition.
But ſeeing that the ratio of AB to AC is given, let the
fame be made of DF to FE, then let DE be drawn.
Therefore the ratio of DF to FE is given. But DF is
given: Therefore (b) FE is given in magnitude. But
the fame FE is alfo given in pofition, and the point F
is given. Therefore (c) the point E is alfo given. But
each of the points D and F is given: Therefore (a)
each of the right-lines DF, FE, and D E is given
in pofition and magnitude. Wherefore (e) the triangle
DEF is given in kind. And feeing that the two triangles
ABC and DEF have an angle equal to an angle, that is
to fay, the angle BAC equal to the angle DFE, and the
fides which conftitute thofe equal angles, proportional;
(f) the triangle ABC is alike to the triangle DEF. But
the triangle DEF is given in kind: Therefore the trian-
gle ABC is given in kind.
PROP.
S 4.
a 29. prop.
b 25. prop.
c 26. prof.
a 29. prop.
b 2. prop.
c 27. prop.
d 26. prop.
e 39. prop.
f 6.6.
280
EUCLIDE'S DATA.
a 2. prop.
b 2. prop.
c 39. prop.
d 5. 6.
a 6. def.
b 2. prop.
£ 19. I.
d 14.5.
PROP. XLII. Plate VIII. Fig. 1, 2.
If the fides of a triangle ABC, are to thofe of another in
given ratio's, the triangle ABC is given in kind,
Conftr. For, let there be expofed the right-line D, gi-
ven in magnitude, and feeing that the ratio of BC to
AC is given, let the fame be made of D to E.
Demonftr. Now D is given, therefore (a) E is alío
given. Again, feeing that the ratio of AC to AB is
given, let the fame be made of E to F. Now E is
given, therefore (b) F is alſo given. Now of three right-
lines, equal to the three given right-lines D, E, and F,
(and of which three lines, two of them, in what man-
ner foever they be taken, are greater than the other,)
let there be conftituted the triangle GHK, in fuch fort
as D may be equal to HK; but È is equal to KG, and
GH, equal to F; therefore each of the faid lines HK, KG
and GH, is given in magnitude: Wherefore (c) the trian-
gle HGK is given in kind, And ſeeing that as BC is to
CA, fo is D to E, and that D is equal to HK, and E
to KG, as BC is to CA, fo HK is to KG. Again,
feeing that as CA is to AB, fo is E to F, and that E
is equal to KG, and F to GH; as CA is to AB, fo is
KG to GH, But it hath been demonftrated, that as
BC is to CA, fo is HK to KG: Therefore in ratio of
equality, as BC is to AB, fo is HK to GH. There-
fore (d) the triangle ABC is alſo given in kind.
PRO P. XLIII. Fig. 3 ·
•
If the fides BC and BA, about one of the acute angles of
a rectangled triangle ABC, have to one another a given
ratio, that triangle is given in kind.
Conftr. Let there be expofed the right-line DE given
in magnitude and pofition, and on it let there be de-
fcribed the femicircle DGE: Therefore (a) the femicircle
DGE is given in pofition.
Demonftr. For the line DE being given, and divided
in two equal parts, the center of the faid circle is given
in pofition, and the femidiameter in magnitude. And
forafmuch as the ratio of BC to BA is given, let the
fame be made of DE to F: Therefore the ratio of DE
to F is given. But DE is given, therefore F (b) is alſo
given. Now BC is greater than (c) AB: Therefore ED
is (d) alfo greater than F. Let DG be fitted equal to F,
and let EG be drawn; then on the center D, with the
diſtance
EUCLIDE'S DAT A.
281
e 6.def.
f 25. prop.
26. prop.
go
h 39. prop•
diſtance DG, let the circle GK be defcribed. Now
that circle (e) is given in pofition, feeing that the center
D is given, and the femidiameter DG is alfo given in
magnitude. But the femicircle DGE is alfo given in
pofition: Therefore (f) the point of interfection G is
given. But the points D and E are alſo given, there-
fore (g) each of the right-lines DE, DG, and EG is
given in pofition and magnitude. Wherefore (b) the
triangle DGE is given in kind. And feeing that the
triangles ABC and DGE have an angle equal to an angle,
to wit, the right angle BAC to the right-angle (i) DĞE,
and the fides about the angles CBA and EDG proporti-
onal. But each of the others ACB and DEG are lefs than
a right-angle: Therefore thofe triangles ABC and DEG
(*) are alike. But the triangle DGE is given in kind: k 7.6.
Therefore the triangle ABC is alſo given in kind.
PRO P. XLIV, Plate VIII. Fig. 4.
If a triangle ABC, hath one angle B given, and that the
fides BA and AC, about another angle BAC, have to one
another a given ratio, the triangle ABC is given in kind.
Conftr. Now the given angle B is either acute or ob-
tufe, (for it was a right-angle in the foregoing propofi-
tion.) Let it be in the firft place acute, and from the
point A let AD be drawn perpendicular to BC.
Demonftr. Therefore the angle ADB is given: But the
angle B is alfo given; and therefore the third angle
BAD is given: Wherefore (a) the triangle ABD is given
in kind; and therefore (b) the ratio of BA to AD is given.
But the ratio of the fame BA to AC is alſo given :
Therefore (c) the ratio of AD to AC is given, and the
angle ADC is a right-angle: Wherefore the triangle (d)
ACD is given in kind: Therefore (e) the angle C is given.
But the angle B is alſo given; and therefore the other
angle BAC is given: Therefore (f) the triangle ABC is
given in kind.
Confir. Now let the angle ABC be obtuſe, and on the
fide CB prolonged, let there be drawn the perpendicular
AD.
Demonftr. Forafmuch as the angle ABC (Fig. 5.) is
given, the angle ABD, which follows it, fhall be given.
But the angle ADB is alſo given: Therefore the third
angle DAB is given. Wherefore (g) the triangle ABD
is given in kind; and therefore (b) the ratio of DA to
AB is given. But the ratio of AB to AC is alfo
i 31. 3.
a 40. prop.
b 3. def.
c 8 prop.
d 43. prop
e 3. def.
f 40. prop
g 40. prop.
h 3. def.
given:
282
EUCLIDE'S DAT A.
i 8. prop.
a 7. prop.
b 3.6.
c 18.5.
given Therefore (i) the ratio of DA to AC is given,
and the angle D is a right-angle. Therefore the triangle
DAC is given in kind, and therefore the angle ACB is
given. But the angle ABC is alfo given: Therefore the
third angle BAC is given. Wherefore the triangle ABC
is given in kind.
PROP. XLV. Plate VIII. Fig. 4.
If a triangle ABC bath one angle BAC given, and that
the line compounded of the two fides AB and AC, about
the jaid given angle BAC, hath to the other fide BC a
given ratio, the triangle ABC is given in kind.
Conftr. For, let the angle BAC be divided into two
equal parts by the line AD, therefore (a) the angle
CAD is given.
Demonftr. Seeing that as AB is to AC, ſo (b) is BD to
CD; by compounding, (c) as the line compounded of
CAB is to CÂ, fo is BC to CD, and by permutation,
as the line compounded of CAB is to CB, fo is CA to
CD. But the ratio of the line compounded of CAB to
BC, is given; therefore the ratio of CA to CD is alſo
d 44. prop. given, and the angle CAD is given. Therefore (d) the
triangle ACD is given in kind, and therefore the angle
Cis given. But the angle BAC is alfo given: There-
fore the third angle B is given: Wherefore (e) the triangle
ABC is given in kind.
e 40. prop.
£ 12. 1.
g 5. I.
h 44. prop.
i 40. prop.
OTHERWISE. Fig. 6.
Conftr. Let BA be prolonged directly unto the point
D, ſo that AI may be equal to AC, and let CD be
joined.
Demonftr. Forafmuch as the ratio of the line com-
pounded of CAB to CB is given, and that AD is equal
to AC, the ratio of the whole line BD to BC is given.
But the angle ADC is alſo given, for it is the half of the
given angle BAC (for the faid angle BAC (ƒ) is equal to
the two internal angles ACD and ADC, which are (g)
equal to one another, becauſe the fides AC and AD are
equal:) Wherefore the triangle BDC (5) is given in kind,
and therefore the angle B is given. But the angle BAC is
alfo given. Therefore the remaining angle ACB is gi-
ven: Wherefore (i) the triangle ABC is given in kind.
PRO P.
EUCLIDE'S DATA.
283
RPO P. XLVI. Plate VIII. Fig. 4.
If a triangle ABC hath one angle B given, and that
the line CAB compounded of the two fides AC and AB
about another angle BAC hath to the other fide BC a
given ratio, the triangle ABC is given in kind.
Conftr. For let the angle BAC be divided into
two equal parts, by the line AD.
Demonftr. Therefore (as hath been fhewn in the fore-
going propofition) the compound line CAB is to CB,
as AB is to BD. But the ratio of the faid compound
line CAB to CB is given: Therefore alfo the ratio of
AB to BD is given. But the angle B is alfo given:
Therefore the triangle ABD (a) is given in kind; and
therefore (b) the angle BAD is given. But the angle
BAC is double to that of BAD; and therefore it is alfo
given. Therefore the third angle C is given. Where-
fore the triangle ABC is given in kind.
OTHERWISE. Fig. 6.
Conftr. Let BA be prolonged directiy, and let AD be
put equal to AC, and let CD be joined.
Demonftr. Forafmuch as the ratio of the line com-
pounded of CAB to CB is given, and that AD is
equal to AC, the ratio of BD to BC is given; and
the angle B is alfo given: Therefore the triangle
CDB (c) is given in kind; and therefore (d) the angle
D is given: Therefore the angle BAC which is double
to BDC, is alfo given: Wherefore the other angle
ACB is given; and therefore the triangle ABC is given
in kind.
PROP. XLVII. Fig. 7.
Retiline figures. AB, CDE, given in kind, are divi-
vided into triangles given in kind.
Conftr. For let the right-lines EB and EC be
drawn.
a 41. prep.
b 2 def.
c 41. prop.
d 3 def.
Demonftr
284
EUCLIDE'S DATA.
a 3. def.
b 41. prop.
c 4. prop,
d 8. prop.
€ 41. prop.
a 3. def.
b 40. prop,
c 80. prop.
d 3. def.
e 8. prop.
f 1.6.
g 41. 1.
h 15.5.
Demonftr. Forafmuch as the rectiline figure ABCDE
is given in kind, the angle (a) BAE is given, and the
ratio of the fide AB to AE is alfo given: Therefore
(b) the triangle BAE is given in kind. Wherefore
the angle ABE is given. But the whole angle ABC
is alfo given: Therefore (c) the remaining angle EBC is
given. But the ratio of the fide AB to the fide
BE, and alſo that of AB to BC is given: There-
fore (d) the ratio of EC to BE is given, and the
angle CBE is alfo given: Therefore (e) the triangle
BCE is given in kind. By the fame way it may be
demonſtrated that the triangle CDE is given in kind.
Therefore rectiline figures given in kind divide them-
felves into triangles given in kind.
PRO P. XLVIII. Plate VIII. Fig. 8.
If on one and the fame_right-line AB, are deſcribed
triangles, as ACB and ABD, given in kind, thofe tri-
angles fhall have to one another a given ratio, as ACB
to ABD.
Conftr. For from the points A and B, let there be
drawn at right-angles on the line AB, the lines AE and
BG, and prolonged unto the points F and H; through
the points C and D, let there be drawn the lines
ECG and FDH, parallel to AB.
Demonftr. Forafmuch then as the triangle ABC is
given in kind, (a) the ratio of CA to BA is given,
and the angle CAB is alfo given; but the angle BAE
is given: Therefore the remaining angle CAE is alſo
given; but the angle CAE is given; and therefore
the other angle ACE is alfo given. Wherefore (b)
the triangle AEC is given in kind. Now the ratio
of EA to AB (c) is given; (for (d) the ratio of EA
to AC, and that of AC to AB is given ;) and in like
manner, the ratio of FA to AB is given. Therefore
(e) the ratio of EA to AF is given; but as AE is
to AF, fo is (f) the parallelogram AH to the paralle-
logram AG; but ACB is (g) the half of AĤ, and
ADB the half of AG; therefore the ratio of the
triangle ACB to the triangle of ADB is given; for
it is the fame ratio with that of AH to AG (½);
that is to fay, of EA to AF, which is given.
PROP
EUCLIDE'S DATA.
285
PROP. XLIX. Plate VIII. Fig. 9.
If on one and the fame right-line AB, there are de-
fcribed any two rectiline figures AECFD and ADB,
given in kind, they shall have to one another a given
ratio (to wit) AECFB to ADB.
Conftr. For let the lines FA and FE be drawn :
Therefore cach of the triangles (a) ABF, AFE, and
ECF is given in kind.
Demonftr. Seeing that on one and the fame right-
line EF there are deſcribed the triangles ECF and EAF,
given in kind; the ratio of ECF to EAF (6) is given.
Therefore by compounding, (c) the ratio of AECF to
EAF is given. But the ratio of the faid EAF to FAB is
given, (d) becauſe they are triangles given in kind, de-
fcribed on one and the fame right-line AF: There-
fore (e) the ratio of AECF to FAB is given. Where-
fore by compounding, (f) the ratio of AECFB to
FAB is given. But the ratio of the fame FAB to
ABD (g) is given: Therefore (b) the ratio of AECFD
to ABD is alſo given.
PROP. L. Fig. 10.
If two right-lines AB and CD, have to one another a
given ratio, and that on those lines there be defcribed recti-
line figures AEB and CFD, alike, and alike pofited,
they will have to one another a given ratio.
Demonftr. To the two lines AB and CD, let there
be taken a third proportional G. Therefore as AB is to
CD, fo is CD to G. But the ratio of AB to CD is gi-
ven: Therefore the ratio of CD to G is alſo given :
Wherefore (a) the ratio of AB to G is given. But (b) as
AB is to G, fo is AEB to CFD: Therefore the ratio of
the fame AEB to CFD is given.
PROP. LI. Fig. 11, 12.
If
two right-lines AB and CD have to one another a
given ratio, and that upon them there be defcribed any reɛti-
line figures AEB and CFD, given in kind, they will have
to one another a given ratio, (to wit, that of AEB to
CFD.)
I
Confir.
a 47. prop.
b 48. prop.
c 6. prop,
d 48. prop.
e 8. prop.
£ 6. prop.
g 48.prop.
h 8. prop.
a 8. prop.
b cor. 19.
20.6.
286
EUCLIDE'S DATA.
a 49. prop.
b 50.prop.
€ 8. prop.
a 29. prop.
b 2. prop.
Conftr. For on AB let the rectangled figure AH be
defcribed alike, and alike pofited to DF.
Demonftr Now DF is given in kind: Therefore alſo
AH is given in kind. But AEB is alfo given in kind,
and deſcribed on the fame line AB: Therefore (a) the
ratio of AEB to AH is given: And feeing that the ra-
tio of AB to CD is given, and that on thoſe lines are de-
ſcribed the rectiline figures AH and DF alike, and alike
pofited, the ratio (b) of the faid line AH to DF is given.
But the ratio of AEB to AH is alſo given: Therefore the
ratio (c) of AEB to DF is given.
PROP. LII. Plate VIII. Fig. 7
If on a right-line BE, given in magnitude, there be de-
fcribed a figure BAE, given in kind, that figure BAE is
given in magnitude.
1
Conftr. For on the fame line BE let the fquare BD be
defcribed. Therefore BD is given in kind * and in
magnitude.
Demonftr. Seeing that on the right-line BE, are de-
ſcribed the two rectiline figures DAE and BD, given in
kind, (a) the ratio of BAE to BD is given: Therefore (b)
BAE is given in magnitude.
Scholium.
* The antient interpreter hath noted here that every
Square is given in kind; for that all the angles thereof are
given; being all equal and right-angles: But alfo the ratio's
of the fides are given; for thofe fides being all equal, their ra-
tio's are alfo equal. Moreover, whenfoever a fquare is expo-
Sed, a fquare equal thereto may be exhibited; and therefore the
Square is given in magnitude, as alfo each fide thereof.
PRO P. LIII. Fig. 13, 14.
If there are two figures AD and EH, given in kind, and
that one fide BD of the one, hath to a fide FH of the other,
a given ratio; the other fides ſhall have alſo to the other fides
given ratio's.
Demonftr.
EUICLDE'S DATA.
287
Demonftr. For feeing that the ratio of BD to FH is gi-
ven, and alfo that (a) of BD to BA, (b) the ratio of the
faid AB to FH is given. But the ratio of the fame FH
to EF (c) is alfo given: Therefore (d) the ratio of AB
to EF is given. In like manner alfo the ratio's of the
other fides to the other fides are given
If
PRO P. LIV. Plate VIII. Fig. 15, 16.
two figures A and B given in kind, have to one another
a given ratio, alfo their fides fall be to one another in a given
ratio
Conftr. For either the figure A is alike and alike pofi-
ted to B, or is not: Let it in the firit place be alike, and
alike pofited; and let there be taken the line G, a third
proportional to the lines CD and EF.
Demonftr. As CD is to G, (a) fo is A to B. But the
ratio of A to B is given; therefore alfo the ratio of CD
to G is given. And feeing that CD, EF, and G, are
proportional, (b) alfo the ratio of CD to EF is given.
But A and B are given in kind: Therefore (c) the
other fides fhall have given ratio's to the other fides.
Now let the figure A be not alike to the figure B,
and let there be defcribed on. EF the figure EH, alike
and alike pofited to A: Therefore the figure EH is given
in kind; but the figure B is alfo given in kind: There-
fore (d) the ratio of B to EH is given; and therefore the
ratio of A to the fame EH (e) is alfo given; But A is
alike to EH: Therefore (by what is abovefaid) the ratio
of CD to EF is given; and in like manner the ratio of
the other fides to the other fides is
a 3. def.
b 8. prop.
c 3. def.
d 8. prop.
a cor. 19.
20. 6.
b 24. prop.
c53. prop.
d 49. prop.
e 8. prop.
OTHERWISE. Fig. 17.
Let
Conftr. Let there be expofed the given line GH; Now
either the figure A is alike to the figure B, or not.
it in the firſt place be alike, and let it be as CD is to EF,
fo is GH to LK; then on GH and LK let the figures M
and N be deſcribed alike, and alike pofited to the faid
A and B, which figures M and N ſhall confequently be
given in kind.
1
}
Demonftr.
1
288
EUCLIDE'S DAT A.
f 22.6.
g 52. prop.
h Sch. 52.
prop.
i sch. 52.
prop.
k 3. prop.
1 53 prop.
a 52. prop.
b
3. prop.
c 54. prop.
d 3. def.
e 2. prop.
Demonftr. Therefore ſeeing that as CD is to EF, fo is
GH to LK, and that on thofe lines CD, EF, GH,
and LK, are defcribed the figures A, B, M, and N,
alike and alike pofited; (f) as A is to B, fo is M to N.
But the ratio of A to B is given: Therefore the ratio of
M to N is given. But (g) M is given, confidering that
it is a figure given in kind, deſcribed on a right-line gi-
ven in magnitude; therefore N is alfo given.
Conftr. 2. Now, on LK let the fquare O be deſcribed:
Therefore (b) the figure O is given in kind.
Demonftr. 2. Wherefore the ratio of O to N is given.
But N is given: Therefore O is given; and confequently
(i) alſo KL. But GH is given: Therefore (k) the ratio
of GH to KL is given. But as GH is to LK, fo is CD to
EF. Therefore the ratio of CD to EF is given; and
therefore the figures A and B being given in kind, (/) the
other fides of the fame figures fhall alſo have to the other
fides given ratio's. But if the figures be not alike, the
latter part of the demonftration here above muſt be ob-
ferved.
PROP. LV. Plate VIII. Fig. 18.
If a Space A be given in kind, and in magnitude, the fide
thereof shall be given in magnitude.
Conftr. For let the right-line BC, given in pofition
and in magnitude, be expofed; and thereon let there be
defcribed the ſpace D, alike and alike pofited to A;
therefore the ſaid ſpace D is given in kind.
Demonftr. For as it is defcribed on the line BC,
given in magnitude, it is alfo (a) given in magnitude.
But the figure A is alfo given: Therefore (b) the ratio
of A to D is given. But thofe figures A and D are given
in kind: Therefore (c) the ratio of the line EF to the
line BC is given. But BC is given: Therefore (d) FF
is alſo given. But the ratio of the fame EF ro FG is
given: Therefore (e) FG is given. And by the fame
ways of reafoning it may be demonftrated that each of
the other fides are given in magnitude.
OTHERWISE. Fig. 19.
Conftr. Let the ſpace GHIKL be given in kind and
in magnitude: I fay that the fides thereof are given in
magnitude. For on the right-line GH let there be de-
fcribed the fquare GM; therefore (ƒ) GM is given in
kind.
Demonftr.
EUCLIDE'S DATA.
289
}
Demonftr. But the ſpace GHIKL is alfo given in kind :
Therefore (g) the ratio of the fame space GK to GM is
given. But GK is given in magnitude: Therefore (h)
GM is alfo given in magnitude; and feeing that GM is
the fquare of the line, GH, (i) that line GH is given in
magnitude. Wherefore in like manner, each of the
other lines HI, IK, KL, and LG, is given.
PROP. LVI. Plate VIII. Fig. 20.
If two equiangled parallelograms A and B, have to one
another a given ratio, as one fide CD of the first A, is to
one fide FG, of the second B; fo the other fide GE, of the
Second B, is to that to which DH, the other fide of the first
A, hath the given ratio that the parallelogram A hath to
the parallelogram B.
Conftr. For let HD be prolonged directly to L, ſo that
as CD is to FG, fo HD may be to DL; and finiſh the
parallelogram DK.
Demonftr. Seeing that as CD is to FG, ſo HD is to
DL, and (a) that CD is equal to KL; as LK is to FG,
fo is GE to DL; and thus the fides about the equal
angles DLK and EGF are reciprocally proportional:
Wherefore (6) DK is equal to B; and therefore feeing
the ratio of A to B is given, and that B is equal to DK,
the ratio of A to DK is given. But as () A is DK
(that is to B) fo is HD to DL: therefore the ratio of
HD to DL is alſo given and feeing that as CD is to
FG, fo GE is to DL, and that the right-line HD hath to
DL a given ratio; to wit, that which the ſpace A hath
to the ſpace B; as CD is to FG, fo GE is to that to
which HD hath the given ratio that the ſpace A hath to
the ſpace B, that is to fay, the ratio of HD to DL.
PROP. LVII. Fig. 21.
If a given Space AD be applied to a given right-line AB
in a given angle CAB, the breadth CA of the application is
given.
Conftr. For on AB, let there be deſcribed the ſquare
AF; therefore (a) the fame AF is given: Let the lines
EA, FB, and CD, be prolonged to the points G and H.
Demonftr. Seeing therefore that each ſpace AD and AF
is given, their ratio is alfo given. But (6) AD is equal
to AH: Therefore the ratio of AF to AH is given:
Wherefore the ratio of EA to AG is given. (For (c) it
T
is
g 49. prop.
h 2. prop.
i sch. 52.
prop.
a 34. 1.
b
b 14.6.
c 1.6.
a ſch. 53.
prop.
b 36. 1.
€ 1.6.
290
EUCLIDE'S DATA.
is the fame with that of AF to AH.) But EA is equal to
AB; therefore the ratio of AB to AG is given, Now
feeing that the angle CAB is given, and the angle GAB
alfo given, the refidue CAG is given. But the angle
CGA is alſo given, being a right-angle: Therefore the
remaining angle ACG is given. Wherefore the triangle
₫ 40. prop. (d) CAG is given in kind: Therefore the ratio of CA to
AG is given. But the ratio of AB to the fame AG is alſo
given: Therefore the ratio of CA to AB is given; and
the faid AB is given: Wherefore CA is alfo given.
a 52. prop.
b 36. 1.
€ 43. I.
d
f
4. prop.
6.
e 24.
55. prop.
$ 34. 1.
h 4. prop.
3. def.
£ 2. prop.
PROP. LVIII. Plate VIII. Fig. 22.
If a given space AB, be applied to a given right-line AC,
wanting by a figure DE, given in kind, the breadths of the
defects are given.
Conftr. For let AC be divided in two equal parts in
the point F: Therefore as well AF as FC is given. On
the faid line FC let there be deſcribed the rectangled
figure FG alike and alike pofited to DE. Therefore
FG is given in kind.
Demonftr. Seeing the figure FG is defcribed on the
right-line FC given in magnitude, the faid rectiline FG
is (a) alfo given in magnitude. But FG is equal to AB
and IL; (for (6) AI and FE being equal, and (c)_FB
and BG alfo equal, the Gnomon ICL is equal to AB;
and therefore their added figure IL common to both,
FG fhall be equal to AB and IL:) Therefore the figures
AB and IL together are given in magnitude. But AB
is given in magnitude: Therefore (d) the remaining
figure IL is alfo given in magnitude. But it is alfo
given in kind, feeing it is (c) alike to DE: Therefore
(f) the fides of the fame IL are given: Wherefore IB
is given; and feeing that it is equal (g) to FD, the ſame
FD is alfo given. But FC is given; therefore the re-
mainder DC () is given; and (i) in a given ratio të
BD, and therefore (k) BD is given.
PROP. LIX. Fig. 23.
If a given Space AB be applied according to a given right-
liue AC, exceeding it by a figure B given in kind, the
breadths of the exceffès CE and CF are given.
Conftr. For DE being divided into two equal parts in
G, let there be defcribed on GE the rectiline figure GH,
alike and alike pofited to CB.
Demonfir..
EUCLIDE'S DAT A.
291
Demonftr. Now feeing that CB is alike to GH, thoſe
figures CB and GH* are about one and the fame diame-
ter, and GH is given in kind, as is CB. But it is de-
fcribed on the given line GE: Therefore (a) the fame
GH is alfo given in magnitude. But AB is given:
Therefore AB and GH are given in magnitude. Now
thofe figures AB and GH, are equal to LI, (for AG, LE,
and EI, being equal, the Gnomon GFH is equal to AB;
and therefore adding GH common to both, LI fhall be
equal to AB and GH;) therefore LI is given in magni-
tude; but is alfo given in kind, fince it is (6) alike to
CB. Therefore (c) the fides of the faid LI are given,
feeing it is equal to GE: Therefore (d) the remainder CF
is given, and in a given ratio (e) to CE. Wherefore (f)
CE is given.
Scholium. Plate VIII. Fig. 24.
* EUCLIDE ſuppoſeth here, that CB and GH are about
one and the fame diameter, but we shall thus demonftrate it :
Let CB and GH be two alike parallelograms difpofed as
above, that is to say, that the equal angles join together in
E, the fide CE meets directly with his homologous fide EH,
and the fide BE, his correfpondent fide EG; and let the di-
ameter FE be drawn, I say that the faid diameter FE pro-
longed, will pass through the point K; that is to ſay, the
parallelograms GH and CB, confift about one and the fame
diameter. For if it be denied, the diameter EF being produ-
ced, will pafs above the point K, or below it, Let it in the
first place pafs above it, and let it cut GK, prolonged in
the point M, and through the point M let there be drawn
MN, parallel to KH, which hall meet EH, prolonged in
the point N, and FB in O.
Demonftr. Forafmuch as the parallelograms GN and
CB are with the parallelogram LO about one and the
fame diameter, they are (g) alike to one another. Where-
fore as FC is to CE, fo is EG to GM. In like manner,
feeing the parallelograms CB and GH are alike, as FC
is to CE, fo is EG to GK: Therefore (b) as EG is to
GM, fo is EG to GK. Wherefore (i) GM and GK are
equal, a part to the whole, which is abfurd: By the
fame way of reaſoning it may be demonftrated. that
the diameter prolonged will not fall below the point
K: Therefore the parallelograms CB and GE confift
about one and the fame diameter.
T 2
PROP.
a 32. prop.
b 24. 6.
c 55. prop.
d 4. prop.
e 3. def.
f 2. prop.
g 24.6:
h 11. 5.
i 9.5.
292
EUCLIDE'S DATA.
@ 55. prop.
b 55. prop
23. def.
b 49. prop.
c 36. I.
d 8. prop.
e 1. 6.
I.
f 8 prop.
PRO P. LX. Plate VIII. Fig. 25.
If a parallelogram AB, given in kind and in magnitude,
be augmented or diminished by a Gnomen CFD, the breadth of
the Gnomon (confifting of the lines CE and DG) are given.
Demonftr. For feeing that AB is given, and the Gno-
mon CFD alfo given, the whole parallelogram BF is
given But it is alfo given in kind, ſeeing it is alike to
BA: Therefore (a) the fides of the fame BF are given;
and therefore each of the lines BE and BG is given.
But each of the lines BC and BD is given; therefore
each of the remaining lines CE and DG is alfo given.
Conftr. Now let the parallelogram BF, given in kind
and in magnitude, be diminished by the given Gnomon
CFD: I fay that each of the lines CE and DG is given.
Demonftr. For ſeeing that BF is given, and the Gnomon
CFD given, the remaining figure AB is alfo given. But
it is alfo given in kind, feeing it is alike to BF: There-
fore (b) the fides of the faid A3 are given, and there-
fore each of the lines CB and BD is given, but each of
the lines BE and BG is given: Therefore alfo each of
the remaining lines CE and DG is given.
PRO P. LXI. Fig. 26.
If to one fide of a figure ABCE, given in kind, there be
applied a parallelogrammic pace CD in a given angle BCF,
and that the given figure AC hath to the parallelogram CD
a given ratio, the parallelogram CD is given in kind.
Conftr. For through the point B, let BH be drawn
parallel to CE, and through the point E let EH be
drawn parallel to CB, and let EC and HB be prolong-
ed to the points K and G.
Demonftr. Forafmuch as the angle BCE is given,
and the ratio of EC to CB, (a) the parallelogram CH
is given * in kind. But the figure ABCE is alfo given
in kind, and is defcribed on the fame line BC, as the
parallelogram CH given in kind is: Therefore (b) the
ratio of the figure ABCE to the parallelograms CH is
given. But by fuppofition the ratio of the faid figure
ABCE to the parallelogram CD is alſo given; and CD
is (c) equal to CG: therefore (d) the ratio of CH to
CG is given. Wherefore the ratio of the line EC to
the line CK is given; (for (e) as CH is to CG, fo is EC
to CK.) But the ratio of EC to CB is alfo given: There-
fore (ƒ) the ratio of the faid CB to CK is given. And
feeing
EUCLIDE'S DATA.
293
feeing that the angle ECB is given, alſo the following
angle BCK (g) is given. But the angle BCF is propofed
given; and therefore the remaining angle FCK is given.
Alfo the angle CKF is given, for that (b) it is equal to
the angle BCK: Therefore the other angle CFK is gi-
ven: Wherefore (2) the triangle FCK is given in kind;
and therefore the ratio of FC to CK is given. But
the ratio of CB to the fame CK is alfo given. There-
fore (k) the ratio of FC to CB is given; and the angle
BCF is alfo given. Wherefore the parallelogram CD
is given in kind.
Scholium.
* Altho' it be manifeft that a parallelogram that hath
one angle given, and the ratio of the fides about the fame
angle alſo given, is given in kind, as Euclide declares, yet
the antient interpreter thus demonftrates it.
Seeing that in the parallelogram CH the angle ECB is
given, the angle CEH is also given; for the right-line EC
falling on the parallels EH and CB, doth make the two
internal angles on the fame part equal to two right-angles.
And therefore ſeeing that the angle ECB is given, the other
angles are given; and feeing that the ratio of EC to CB is
given, and that BH is equal to CE, and EH to BC, the
ratio of the fides to one another is also given.
PROP. LXII. Plate VIII. Fig. 11, 12.
If two right-lines AB and CD, have to one another a
given ratio, and that on one of them AB, there be deſcribed
a figure AEB, given in kind; but on the other CD, a paral-
lelogrammic Space DF in a given angle DCF, and that the
figure AEB bath to the parallelogram DF a given ratio,
the parallelogram DF is given in kind.
Conftr. For on the line AB let there be defcribed the
parallelogram AH, alike and alike pofited to DF.
Demonftr. Seeing that the ratio of AB to CD is given,
and that on thofe lines are deſcribed the rectiline figures
AH and FD, alike and alike pofited, (a) the ratio of AH
to FD is given. But the ratio of FD to AEB is alſo
given: Therefore (6) the ratio of AH to AEB is given.
But the angle ABH is alfo given, being equal to the
angle FCD, and fo the figure AEB is given in kind;
and to AB one of the fides thereof, the parallelogram
$13. 1
4. prop.
h 29. I.
i 40. prop.
k 8. prop.
a 50. prop.
b 8. prop.
T 3
AH
294
EUCLIDE'S DATA.
c 61. prop.
a 49. prop.
J
2 12. 2.
b 40. prop.
c 3. def.
d I. 6.
I.
e 41. 1.
£ 8. prop.
AH is applied in a given angle ABH, and the ratio of
the faid figure AEB to the faid parallelogram AH is gi-
ven: Therefore (c) the parallelogram AH is given in
kind; and therefore FD which is alike thereto, is alfo
given in kind.
PRO P. LXIII. Plate VIII. Fig. 27.
If a triangle ABC be given in kind, the ſquares BE, CD,
and CF, which is defcribed on each of the fides, fhall have
a given ratio to the triangle ABC.
Demonftr. For feeing that on one and the fame right-
line BC, there are defcribed the two rectiline figures
ABC and CD, given in kind, (a) the ratio of the fame
ABC to CD is given; and therefore the ratio of the
fquares BE and CF, to the triangle ABC is alſo given.
PROR. LXIV. Fig. 5.
If a triangle ABC, hath an obtufe angle ABC given,
that space by which the fide AC fubtending the obtufe angle
ABC, is more in power than the fides AB and BC, that
comprehend the faid angle, shall have a given ratio to the
triangle ABC.
Conftr. Let the line CB be prolonged directly, and
from the point A let the perpendicular AD be drawn :
I fay that the ſpace by which the fquare of the line AC
doth exceed the fquares of the lines AB and BC, that is
to fay, (a) the double of the rectangle contained under
CB and BD, fhall have a given ratio to the triangle
ABC.
Demonftr. For feeing that the angle ABC is given,
the angle ABD is alfo given; but the angle ADB
is alfo given; therefore the other angle BAD is
given: Wherefore (b) the triangle ABD is given in
kind; therefore (c) the ratio of AD to DB is given.
But as AD to DB, fo (d) the rectangle of AD and BC is
to the rectangle of BC and BD. But the ratio of AD to
BD is given: Therefore alfo is the ratio of the rectangle
of AD and BC to the rectangle of BC and BD given:
Wherefore the ratio of the double of the faid rectangle
BC and BD to the rectangle of AD and BC is alfo given,
But the faid rectangle of AD and BC hath alſo a given
ratio to the triangle ABC (to wit, a double ratio; for
the rectangle is (e) double to the triangle) therefore the
ratio of the double of the rectangle of BC and BD (ƒ)
to
13
Plate VII. Facing Pag. 294.
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EUCLIDE'S DATA.
295
to the triangle ABC is given. But the fame double of
the rectangle of CB and BD is that ſpace by which the
fquare of the line AC doth exceed the fquares of the lines
ÁB and BC: Therefore the fame ſpace hath a given ra-
tio to the triangle ABC.
PRO P. LXV.
Plate VIII. Fig. 4.
If a triangle ABC, hath one acute angle ACB given,
that space, by which the fide fubtending the ſaid acute angle
is lefs in power than the fides comprehending the fame acute
angle, ſhall have a given ratio to the triangle.
Conftr. From the point A let there be drawn the line.
AD, perpendicular to BC: I fay, that ſpace by which
the fquare of the line AB is less than the fquares of the
lines AC and CB, that is to fay, (a) the double of the
rectangle of BC and CD, hath a given ratio to the
triangle ABC.
Demonftr. For feeing that the angle C is given, and
the angle ADC alfo given, the other angle DAC is
given: Wherefore the triangle (b) ADC is given in kind;
and therefore the ratio of AD to DC is given, and con-
fequently alfo (c) that of the rectangle of BC and CD to
the rectangle of BC and AD: Therefore the ratio of the
double of the rectangle of BC and CD to the rectangle of
BC and AD is given. But the ratio of the fame rect-
angle of BC and AD to the triangle ABC is given (for
(d) the rectangle is double to the triangle :) Therefore (c)
the ratio of the double of the rectangle of BC and
CD to the triangle ABC is given. And feeing that
the fame double of the rectangle of BC and CD is that
whereby the fquare of the line AB is lefs than the fquares
of the lines AC and BC, that ſpace by which the ſquare
of the line AB is lefs than the fquares of the lines AC and
BC, ſhall have a given ratio to the triangle ABC.
PRO P. LXVI. Fig. 4.
If a triangle ACB, hath one angle B given, the rectangle
made of the lines AB and BC, containing the fame angle,
ſhall have a given ratio in the triangle.
Conftr. For from the point A let AD be drawn per-
pendicular to CB.
Demonftr. Therefore ſeeing that the angle B is given,
and alſo the angle ADB; the other an le BAD is like-
wife given. Wherefore the triangle ADB (a) is given
a 13. 2.
b 40. prop.
C 1.6.
d 41. 1.
e 8. prop.
a 40. prep.
T 4
in
296
EUCLIDE'S DAT A.
b1.6.
€ 41. prop.
a 8. prop•
a 4. 6, &
14.5.
b 40. prop.
c 50. prop.
dz. 6.
e 1. 6.
£ 1.6.
in kind; and confequently the ratio of AB to AD is
given. But as AB is to AD, (b) fo the rectangle_of
AC and CB is to the rectangle of CB and AD:
Therefore the ratio of the rectangle of AC and CB to
the rectangle of CB and AD is given. But the ratio of
the faid rectangle of CB and AD to the triangle ACB
is alfo given; (for that it is double ratio, the rectangle
being double (c) to the triangle :) Therefore (d) the ratio
of the rectangle of AC and CB to the triangle ABD
is given.
PRO P. LXVII. Plate VIII. Fig 28.
If a triangle ABC hath one angle BAC given, that ſpace
by which the ſquare of the line compounded of the two fides
BA and AC, that contain the fame given angle BAC doth
exceed the fquare of the other fide, fhall have a given ratio
to the triangle ABC.
Conftr. For let BA be prolonged in fuch fort as that
AD may be equal to AC, then having drawn DCE
infinitely, from the point B let BE be drawn parallel to
AC, meeting the faid DE in the point E.
Demonftr. Forafmuch as AD is equal to AC, (a) DB is
equal to BE; (for the two triangles ADC and BDE are
alike) and from the top B is drawn to the baſe DE, the
right-line BC: Therefore the rectangle of DC and CE,
with the fquare of BC, is equal to the fquare of BD ; but
the fame BD is compounded of BA and AC; therefore
the fquare of the compound of AB and AC is greater than
the fquare of BC, of the rectangle of DC and CE.
Now I fay that the rectangle of DC and CE hath a
given ratio to the triangle ABC: Forafmuch as the angle
BAC is given, the angle DAC is alfo given. But each
of the angle ADC and ACD is given, it being the half
of the angles BAC which is given. Therefore (b) the
triangle ADC is given in kind; and therefore the ratio
of DA to DC is given. Therefore (c) the ratio of the
fquare of the faid DA to the fquare of DC is alfo given.
And ſeeing that as BA is to ÂD, (d) fo is EC to CD,
and alfo as BA is to AD, (e) fo is the rectangle of BA
and AD to the fquare of AD; and as EC is to CD, (f)
fo alfo is the rectangle of EC and CD to the fquare of
CD; by permutation, as the rectangle of BA and AD
is to the rectangle of EC and CD, fo is the fquare of
AD to the fquare of C. But the ratio of the faid
fquare of AD to the fquare of DC is given: Therefore
the
}
EUCLIDE'S DATA.
297
the ratio of the rectangle of BA and AD to the rectan-
gle of EC and CD is alfo given. But AD is equal to
AC: Therefore the ratio of the rectangle of BA and
AC to the rectangle of EC and CD is given. But the
ratio of the rectangle of BA and AC to the triangle
ABC (g) is given, becauſe the angle BAC is given:
Therefore (b) the ratio of the rectangle EC and CD to
the triangle ABC is given. But the rectangle of EC
and CD is that whereof the fquare of the line compound-
ed of BA and AC is greater than the fquare of BC:
Therefore that ſpace by which the fquare of the line
compounded of BA and AC is greater than the fquare
of BC, fhall have a given ratio to the triangle ABC.
Scholium.
* EUCLIDE fuppofeth in this place, that when in an
Ifofceles triangle a right-line is drawn from the top to the
bafe, the fquare of that line, with the rectangle contained
under the fegments of the bajes, is equal to the fquare of
either of the other legs, which the antient interpreter doth
thus demonftrate.
Conftr. Let ABC be an Ifofceles triangle, whofe legs
are AB and AC; and from the top A let AD be drawn
to the baſe BC: I fay, that the fquare of AD with the
rectangle of BD) and DC, is equal to the fquare of either
of the legs AB or AC.
Demonftr. Now the line AD (Fig. 4.) is perpendicular
to BD, or not: Let it in the first place be perpendicular:
Therefore it will cut the bafe BC into two equal parts
in the point D; and therefore the rectangle contained
under BD and DC is equal to the fquare of the ſaid BD,
and adding to them the common fquare of AD, the
rectangle of BD and DC with the fquare of AD, ſhall
be equal to the fquares DB and AD. But to thoſe
fquares of AD and DB (i) the fquare AB is equal:
Therefore the fquare of AB is equal to the rectangle of
BD and DC, and the ſquare of AD together.
go
66. prop.
h 8. prop.
i 47. 1.
Now fuppofe AD not to be perpendicular, but that
from the point A there doth fall on BC the perpendicu-
lar AE, that being fo, BC fhall be cut into two parts
equally in the point E, and unequally in D. Wherefore
the rectangle of BD and DC, with the fquare of DE, (k) k 5. 2.
is equal to the fquare of BE; and adding the common
fquare of AE, the rectangle of BD and DC, with the
fquares of DE and AE, fhall be equal to the fquares of
BE
298
EUCLIDE'S DATA.
1 47. I.
m 8. prop.
n 1.6.
0 41. I.
P 37. I.
q 8, prop.
I 47. I.
$ 5. 2,
BE and AE. But (1) the fquare AD is equal to the two
fquares of DE and AE: Therefore the rectangle of BD
and DC, with the fquare of AD, is equal to the ſquares
of BE and AE. But to theſe ſquares of BE and AE
the fquare of AB is equal: Therefore the fquare of AD,
with the rectangle of BD and DC, is equal to the fquare
of AB.
OTHERWISE.
Conftr. Having done, as in the foregoing Demonſtra-
tion, from the point A, (Fig. 29.) let AF be drawn
perpendicular to CD, and let AE be drawn.
Demonftr. Forafmuch as the angle BAC is given, the
half thereof ACF ſhall be alfo given. But the angle
AFC is given; and therefore the triangle AFC is given
in kind: Therefore the ratio of AF to FC is given.
But the ratio of CD to the fame FC is alſo given, ſeeing
that CD is double to FC: Therefore (m) the ratio of CD
to AF is given; and therefore alfo the ratio of the
rectangle of CD and EC, to the rectangle of AF and
EC, is given; (for it is the fame ratio (2) as that of CD
to AF.) But the ratio of the rectangle of AF and FC
to the triangle ACE is given; feeing it is double (o) to
the fame triangle. Therefore the ratio of the rectangle
of CD and CE to the triangle ACE is alfo given. But
the triangle ACE is equal to the triangle ABC (p), they
being both conftituted on one and the fame baſe AC,
and between the fame parallels AC and BE: Therefore
(9) the ratio of the rectangle of CE and CD to the
triangle ABC is given. But the faid rectangle of CE
and CD is the ſpace by which the fquare of the line
compounded of AB and AC, is greater than the fquare
of BC: Therefore that ſpace by which the fquare of
the line compounded of AB and AC is greater than the
fquare of BC, hath a given ratio to the triangle ABC.
OTHERWISE. Fig. 4.
For the given angle A is either a right, acute, or
obtufe angle: Let it in the firft place be fuppoſed a
right-angle Therefore the fquare of the line com-
pounded of BAC, is greater than the fquare of BC, by
twice the rectangle of BA and AC; (feeing that () the
fquare of BC is equal to the fquares of BA and AC;
and the fquare of the line compounded of BAC (s) is
equal
EUCLIDE'S DATA.
299
equal to thoſe two fquares of BA and AC, and twice
the rectangle of the faid BA and AC :) Wherefore the
ratio of double the rectangle of BA and AC to the
triangle ABC is given.
Conftr. Now let the angle C (Fig. 4.) be fuppofed
acute, and from the point A let there be drawn on CB
the perpendicular AD.
u 4. 2.
2.
Demonftr. Forafmuch as the triangle CAB is an Oxi-
gonium triangle, and the perpendicular AD being drawn,
the fquares of CA and CB are equal (†) to the ſquare of t 13. 5.
AB with twice the rectangle of CB and CD; adding
therefore the common double rectangle of CA and CB,
the fquares of CA and CB, with the double rectangle
of the faid CA and CB, that is to fay, (u) the
fquare of the line compounded of ACB, are equal to
the fquare of AB, with the double of the rectangle of
CD and CB, and over and above the double of the
rectangle of AC and CB, that is to fay, the double of
the rectangle contained under the compound-line of ACD
and CB (for the rectangle of ACD and CB is (x) equal x 1.
to the rectangles of AC and CB, and of CD and CB :)
Therefore the fquare of the line compounded of ACB is
greater than the ſquare of AC, by double the rectangle of
ACD and CB. And feeing that the angle ACB is given,
and the angle BDA alfo given, the other angle CAD is
given: Therefore (y) the triangle CAD is given in
kind, and therefore the ratio of CD to CA is given,
and by confequence the ratio of the line compounded of
ACD to CA (≈) is alfo given. Wherefore the ratio of
the rectangle of thofe lines compounded of ACD and
CB (a) to the rectangle of AC and CB is alfo given.
But the ratio of the ſaid rectangle of AC and CB to the
triangle CAB (6) is given, feeing the angle C is given; b 66. prop.
therefore the ratio of double the rectangle of the line
compounded of ACD and CB to the triangle CAB
is given.
Laftly, let the angle BAC (Fig. 30.) be ſuppoſed to be
obtufe, and having prolonged BA, from the point C,
let the perpendicular CE be drawn on the faid line BA
prolonged; and let AF be propoſed to be equal to AE.
Demonftr. Forafmuch as the angle BAC is obtufe, and
the perpendicular CE being drawn, the fquares of AB
and AC, and the double of the rectangle under BA and
AE, or AF, are all alike equal (c) to the fquare of BC,
and adding the common double rectangle of BA and AC,
the fquares of the faid AB and AC, with the double of
+
the
y 40. prop.
z 6. prop.
a I. 6.
C 12. 2,
300
EUCLIDE'S DATA.
d 4. 2.
e 1. 2.
£ 13. 11.
g 40. prop.
the rectangle of the fame AB and AC, that is to fay.
(d) the fquare of the line compounded of BAC and the
double of the rectangle of BA and AF are together
equal to the fquare of BC, with the double of the rect-
angle of BA and AC. Let the common double of the rect-
angle of BA and AF be taken away, and there will re-
main the fquare of the line compounded of BAC, equal
to the fquare of BC, with the rectangle of AB and CF;
(for the rectangle of AB and AC is equal (e) to the two
rectangles of AB and AE, and of AB and CF :) There-
fore the fquare of the line compounded of BAC is greater
than the fquare of BC by the double of the rectangle
of AB and CF. And forafmuch as the angle BAC is
given, the angle CAE (ƒ) is given. But the angle AEC
is alfo given; therefore the other angle ACE is given:
Wherefore (g) the triangle ACE is given in kind, and
therefore the ratio of CA to AE, that is to ſay, to AF
is given. Therefore (b) the ratio of the faid CA to FG is
alſo given. But the ratio of the fame CA to CE is
given; therefore (i) the ratio of CE to CF is alfo given.
Wherefore the ratio of the rectangle of EC and AB to
the rectangle of FC and AB is given; (for the rectangle
is to the rectangle (k) as CE is to CF) and alſo that of the
rectangle of AC and AB to the rectangle of EC and
AB. Therefore (/) the ratio of the rectangle of FC and
AB to the rectangle of AC and AB is given. But the
ratio of the rectangle of AC and AB to the triangle ABC
m 66. prop. (m) is given: Therefore alfo the ratio of the double of
the rectangle of FC and AB, to the triangle ABC is given.
But the fame double of the rectangle of FC and AB, is
that whereby the fquare of the line compounded of BAC
is greater than the fquare of BC, wherefore that ſpace
by which the fquare of the line compounded of BAC is
greater than the fquare of BC, hath a given ratio to the
triangle ABC.
h 5. prop.
i 8. prop.
k 2. 6.
18. prop.
OTHERWISE. Plate VIII. Fig. 31.
Conftr. Let the line BA be prolonged to the point D, in
fuch fort as AD may be equal to AC, and let CD be drawn.
Demonftr. Forafmuch as the angle BAC is given, each
of the angles ADC and ACD, which is the half thereof,
ſhall be alfo given; and therefore the other angle DAC
n 40. prop. is alfo given: Therefore (2) the triangle ACD is given in
kind. Wherefore the ratio of AC to CD is given. And
forafmuch as the angle ADC is given: Let each of the
angles
1
EUICLDE'S DATA.
301
1
0 4. 6.
P 5. 2.
9 5. 2.
r 17.6.
angles DEC and AFC be made + equal to the faid
ADC: Therefore feeing that the angle BDC is equal
to the angle DEC, and the angle DBE is common to
the triangles DBE and DBC, the other angle BDE is
equal to the other angle BCD; and therefore the trian-
gle BDE is equiangled to the triangle BDC. Therefore
() as EB is to BD, fo is BD to CB: Wherefore the rect-
angle of EB and CB, that is to fay, (p) the rectangle of
EČ and CB, (9) with the ſquare of CB is equal, (†) to
the fquare of BD, that is to fay, to the fquare of the line
compounded of BAC; for AD is equal to AC; and
therefore the rectangle of EC and CB with the fquare
of CB, that is to fay, the fquare of the line compounded
of BAC is greater than the fquare of the rectangle of
BC and CE: I fay therefore that the ratio of the faid
rectangle of BC and CE to the triangle ABC is given.
Forafmuch as the angle BDE is equal to the angle
BCD, and the angle ADC equal to the angle ACD,
the other angle CDE is equal to the other angle ACB:
But the angle DEC is equal to the angle AFC; therefore
the remaining angle CAF is equal to the remaining
angle DCE. Wherefore the triangle AFC is equian-
gled to the triangle DCE; and therefore (s) as CA is s 4. 6.
to AF, fo is CD to CE; and by permutation, as ACis
to CD, fo is AF to CE. But the ratio of AC to CD
is given: Therefore alfo the ratio of AF to CE is given.
From the point A let AH be drawn perpendicular to BC:
Forafmuch as the angle AFC is given, and the angle
AHF alfo given, the third angle HAF is given:
Wherefore () the triangle AHF is given in kind; and
by confequence the ratio of AF to AH is given. But
the ratio of AF to CE, is alfo given: Therefore (u)
the ratio of AH to CE is given; and therefore the
ratio of the rectangle of AH and BC (x) to the rectangle
of BC and CE is alfo given. But the ratio of the
rectangle of AH and BC, to the triangle ABC is like-
wife given; (for the rectangle (y) is double to the tri-
angle) and the rectangle of BC and CE is that whereby
the fquare of the line compounded of BAC is greater
than the fquare of BC. Therefore that space by which
the fquare of the line compounded of BAC is greater
than the fquare of BC has a given ratio to the triangle
ADC.
t 40. prop.
u S. prop.
X I. 6.
1
y 41. I.
Scholium.
302
EUCLIDE'S DATA.
£ 29. I.
a 1. 6.
b 14. 6.
Scholium.
† The antient Interpreter pretending to shew the con=
fruction of the angle DEB equal to the angle ADC, faith
that on the line BD and in the point D, the angle BDE
ought be made equal to the angle BCD, and that the right-
lines BC and DE be drawn until they interfect in E, in fuch
fort as he fuppofeth the angle BCD, to be given, but it is not.
The fame Interpreter afterwards Thews how there
may univerfally from a given point be drawn a right-
line, given in pofition to a right-line, making an angle
equal to a given angle: But we will aljo reject this
way, feeing we have elsewhere fhewn another more brief
and easy. For example, if we would from the point D
draw to the line BC given in pofition a right-line, making
an angle equal to a given angle ADC, as is here required,
we have no more to do but to affume the point K in the
faid line BC, and there make the triangle CKL equal to
the given angle ADC: If the line KL doth meet with the
point D, it ſhall be the line required. But if it meet not
with it, from the point D let there be drawn the line DE
parallel to the faid KL, cutting BC prolonged in E, and
the angle DEC fhall be equal to the given angle ADC, for
on the two parallel-lines, LK and DE, there doth fall the
line BE; and therefore the angle DEC (z) is equal to the
angle LKC, which hath been made equal to the given
angle ADC; and by confequence the fame angle DEC is alſo
equal to ADC,
PRO P. LXVIII. Plate VIII. Fig. 32, 35.
If two parallelograms AB and CD have to one another
a given ratio, and that a fide hath alſo a given ratio to a
fide, the other fide ſhall have likewife a given ratio to the
other fide.
Conftr. Let the ratio of BE to FD be given: I fay
the ratio of AE to FC is alſo given. For to the right-
line EB let there be applied the parallelogram EH, equal
to the parallelogram CD, and conftituted in fuch fort
as AE and EG may make one right-line: † Therefore
KB and BH will alfo make one right-line.
Demonftr. Forafmuch as the ratio of AB to CD is
given, and that EH is equal to the faid CD; the ratio
of (a) AB to EH is given; and therefore the ratio of AE
to EG is alfo given. Seeing therefore that EH is equal
and
EUCLIDE'S DATA.
303
and equiangled to CD, as (b) EB is to FD, fo is FC
to EG. But the ratio of EB to FD is given: There-
fore alfo the ratio of FC to EG is given. But the
ratio of AE to the fame EG is alfo given: Therefore
the ratio of AE to FC is given.
Scholium.
d 29. 1.
† EUCLIDE having pofited AE and EG directly in one
right-line, prefently concludes that KB and BH jhall alſo
make a right-line; but we ball demonftrate it thus Seeing
the lines ÅE and EG are pofited directly, the angles AEB
and BEG (c) are equal to two right-angles; and feeing c 13. 1.
that AB is a parallelogram, the lines AK and EB are
parallels, on which the line AE doth fall; and therefore
the two internal angles A and BEA (d) are also equal to
trvo right-angles, and taking away the common angle BEA,
there will remain the angle A, equal to the angle BEG;
and confequently their oppofite angles EBK and H are alſo
equal to one another. Again, feeing that BG is a paral-
lelogram, the two lines BE and HG are parallels, on which
BH falls; and therefore the two internal angles H
and ЕBH are equal to two right-angles. But it hath been
demonftrated that H is equal to EBK: Therefore the two
angles EBK and EBH are alfo equal to two right-angles;
and therefore (e) the tavo lines KB and BH meet di-
rectly according to EUCLIDE.
OTHERWISE. Plate VIII. Fig. 34, 35.
Conftr. Let the given right-line K be expofed, and
feeing that the ratio of A to B is given, let the fame
be made of K to L; therefore the ratio of K to L is
alfo given.
e 14. Is
I
g 1. prop,
h 23.6.
Demonftr. But K is given; therefore (f) L is alfo given. f 2. prop. }
Again, feeing that the ratio of CD to EF is given, let
the fame be made of K to M: Therefore the ratio of K
to M is given. But K is given, therefore (g) M is alſo
given; and therefore the ratio of L to M is given.
Now feeing that A is equiangled to B, (b) the ratio
of the faid A to B is compounded of that of the fides,
that is to fay, CD to EF, and of CG to EH. But
alfo the ratio of K to L is compounded of K to M,
and of M to L; therefore the ratio compounded of
CD to EF, and of CG to EH, is the fame with that
which is compounded of K to M, and of M to L (the
ratio
304
EUCLIDE'S DATA.
a 40. prop.
b35. prop.
c 68. prop,
d 29. 1.
e 29. 1.
f 34. 1.
ratio of K to L being the fame as of A to B :) But
the ratio of CD to EF is the fame as of K to M: There-
fore the other ratio of CG to EH is alfo the fame as of
M to L. But the faid ratio of M to L is given: There-
fore alſo the ratio of CG to EH is given.
PRO P. LXIX. Plate VIII. Fig. 36, 37.
If trvo parallelograms, CB and EH, having the angles
D and F given, and that a fide hath alfo a given ratio to
a fide; in like manner the other fide shall have a given
ratio to the other fide.
Conftr. Let the ratio of BD to FH be alfo given: I
fay that the ratio of AB to EF is given. For if CB be
equiangled to HE, it is manifeft by the precedent Pro-
pofition; but if it be not equiangled thereto, let the
right-line DB be conftituted, and in the given point
B therein, let the angle DBK be made equal to the
angle EFH, and finish the parallelogram DK.
Demonftr. Forafmuch as each of the angles BKL
and BAK is given, † the other angle KBA is given :
Wherefore the triangle (a) ABK is given in kind; and
therefore the ratio of AB to BK is given. But the
ratio of CB to EH is fuppofed to be given, and (b) CB
is equal to DK; therefore the ratio of DK to EH is
given; and feeing that DK is equiangled to EH, and
the ratio of the faid DK to EH is given, as alfo that
of DB to FH, (c) the ratio of BK to FE is given. But
the ratio of the faid BK to BA is alſo given: There-
fore (d) the ratio of AB to FE is given.
Scholium.
+ EUCLIDE fuppofeth here, that a parallelogram having
one angle given, all the other angles are aljo given, and
as well the antient Interpreters as others give the rea-
fons why, the angle F being given, the other angle E ball
be alfo given, it being the remainder of two right-angles,
for that on the parallel-lines EG and FH there falls the
line EF, which makes (e) the two internal angles (of the
Jame part) F and G, equal to two right-angles. But to
thofe angles (f) the oppofite angles Gˇand H are equal,
and therefore they are alfo given.
From whence it follows that the angles BDC and F
being given by fuppofition, all the other angles of the two
parallelograms CB and EH, are alfo given: Therefore the
angle DBK having been made equal to the angle F, the
angle K fhall be equal to the angle E, and given as that
is :
EUCLIDE'S DATA. 3
305
is; But the angle BAL, which is oppofite to the given angle
BDC, is also given; and therefore BAK, which is the re-
mainder of two right-angles, jhall be alf given; in luch
fort as in the triangle ABK, the two angles BAK and
BKA are given, as EUCLIDE declars in this place.
PROP. LXX. Plate VIII. Fig. 38, 39.
If of two parallelograms AB and EH, the fides about
the equal angles, or about the unequal angles (yet never-
theless given angles) have to one another a given ratio,
to it, (AC to EF, and CB to FH) alfo the fame paralle-
lograms AB and EH hall have to one another a given
ratio.
Conftr. For let AB be prolonged to EH, and on the
right-line CB let the parallelogram CM be applied equal
to the parallelogram EH, in fuch fort as AC may be
directly to CN; that is to fay, that AC and CN make
one right-line; and by confequence DB ſhall be (a) di-
rectly with BM.
Demonftr. Forafmuch then as CM is equiangled and
equal to EH, the fides about the equal angles fhall be
reciprocally (b) porportional: Wherefore as BC is to HF,
fo is FE to NC. But the ratio of BC to HF is
given,
Therefore the ratio of FE to NC is alfo given.
But the ratio of AC to the fame EF is given:
Therefore (c) the ratio of AC to NC is alfo given.
Wherefore the ratio of AB to CM is given: (for it is,
the fame (d) as of AC to CN.) But CM is equal to EH :
Therefore the ratio of AB to EH is given.
Conftr. Now fuppofe AB not to be equiangled to EH,
and on the right-line CB, and in the given point C
therein, let there be conftituted the angle BCK, equal
to the given angle F, and fo finish the parallelogram
CL.
Demonftr. Forafmuch as the angle ACB is given,
and the angle BCK alfo given, the remaining angle
ACK is given: Therefore the triangle ACK (e) is given
in kind: and therefore the ratio of AC to CK is given:
But the ratio of AC to EF is alſo given; Therefore the
ratio of CK to EF is given, But the ratio of BC to HF
is alfo given, and the angle BCK is equal to the angle
F; therefore (by the first part of this propofition) the
ratio of CL to EH is given. But to the faid CL, AB is
equal: Therefore the ratio of AB to EH is given.
U
PROP.
a ſch. 68.
prop,
b 14. 6.
c 8. prop.
d 1. 6
e 40. prop.
306
EUCLIDE'S DATA.
$ 70. prop.
b 34. prop.
@ 29. 1.
PROP. LXXI. Plate VIII. Fig. 40, 41.
If of two triangles ABC and DEF, the fides about
the equal angles A and D, or else about the unequal an-
gles (yet nevertheless given angles) have to one another a
given ratio (to wit, AB to DE, and AC to DF) the fame
triangles fhall have alfo to one another
a given ratio
Conftr. Let the parallelograms AG and DH be
finiſhed.
ABC to DEF.
Demonftr. Seeing that the two parallelograms AG and
DH, have the fides about the equal angles A and D, or
elfe about the unequal angles (nevertheleſs given) in
a given ratio to one another, the ratio (a) of the paral-
lelogram AG to the parallelogram DH is given. But
the triangle ABC is the half of the parallelogram AG (b)
and the triangle DEF the half of the parallelogram DH.
Therefore the ratio of the triangle ABC to the triangle
DEF is given.
PRO P. LXXII. Fig. 42, 43.
If of two triangles ABC and DEF, the bafes BC and
EF, are in a given ratio, BC to EF, and that from the angles
A and D, there be drawn to those bajes the right-lines AG
and DH, making the angles AGC and DHF equal, or elfe
unequal (yet nevertheless given) which shall have to one ano-
ther given ratio's AG to DH, thofe triangles ABC and
DEF ſhall have alſo a given ratio to one another, to wit,
ABC to DEF.
Conftr. For let the parallelograms KC and LF be
finiſhed.
Demonftr. Forafmuch as the angles AGC and DHF
are equal, or unequal (yet given) and that the angle
AGC (a) is equal to the angle KBC, and alſo the angle
DHF equal to the angle LEF, the angles at the points B
and E are equal, or elſe unequal (yet given,) and becauſe
the ratio of AG to DH is given, and AG is equal to
KB, and DH is equal to LE, therefore the ratio of KB to
LE is given. But the ratio of BC to EF is alſo given,
and the angles at the points B and E are equal, or elfe
b 70 prop. unequal (yet given :) Therefore (6) the ratio of the
paralel-
EUCLIDE'S DATA.
307
parallelogram KC to the parallelogram LF is given ; and
therefore the ratio of the triangle ABC to the triangle
DEF is given, feeing thoſe triangles (c) are the one half
of the parallelograms.
PRO P. LXXIII. Plate VIII. Fig. 38, 39.
If of two parallelograms AB and EH, the fides about
the equal angles C and F, or else about the unequal angles
(but nevertheless given) are in fuch fort to one another, that
as the fide CB of the firft, is to the fide EH of the fecond; fo
the other fide EF of the fecond, is to fome other right-line CN.
But that the other fide AC, hath alfo to the fame right-line
CN a given ratio, thofe parallelograms will have alſo to one
another a given ratio AB to EH.
Conftr. For in the firft place, let the parallelogram
AB be equiangled to EH, and having placed CN directly
to AC: Let the parallelogram CM be finifhed.
Demonft. Forafmuch then as CB or NM its equal, is
to FH, fo is EF to CN, and that the angles N and F
are equal (for N is equal to the angle AČB, which is
put equal to F) the parallelograms CM and EH (a) are
equal: But as AC to CN, ſo (b) the parallelogram AB is
to the parallelogram CM or EH: Therefore feeing that
the ratio of AC to CN is given, the ratio of AB to EH
is alfo given.
Conftr. 2. Now fuppofe the parallelogram AB not to
be equiangled to the parallelogram EH, and let there be
conftituted at the given point C in the line CB, the angle
BCK, equal to the angle EFH, and fo finish the paral-
lelogram CL.
:
Demonftr. 2. Seeing that each of the angles ACB and
KCB is given, the remaining angle ACK is alſo given.
But (c) the angle CAK is given, as alfo the remaining an-
gle AKC: Therefore (d) the triangle ACK is given in
kind and therefore the ratio of AC to CK is given.
But the ratio of the fame AC to CN is alfo given :
Therefore (e) the ratio of CK to CN is given. And fee-
ing that as CB is to FG; fo is EF to the right-line CN, to
which the other fide KC hath a given ratio, and that the
angle BCK is equal to the angle F, the ratio of the
parallelogram CL to the parallelogram EH is given (by
the first part of this propofition) but the parallelogram
CL is equal to the parallelogram AB: Therefore the
ratio of the parallelogram AB to the parallelogram EH
is given.
U z
PROP
€ 41. I.
a 14.6.
b 1.6.
ċ ſch. 69.
prop:
d 40. prop.
e 8. prop:
308
EUCLIDE'S DAT A.
*
■ ſch. 68.
prop.
b 1.6.
€ 14. 6.
d 36. 1.
e fch. 69.
prop.
PRO P. LXXIV. Plate VIII. Fig. 38, 39.
If two parallelograms AB and EH, in equal angles
C and F, or elſe in unequal angles (yet nevertheless gi-
ven angles) have a given ratio to one another, as one fide
CB of the first shall be to one fide FH of the fecond, fo
the other fide EF of the fecond, fhall be to that to the
which the other fide AC of the first hath a given ratio.
Conftr. For either AB is equiangled or not; fuppofe
it in the firſt place to be equiangled, and to the right-line
BC let there be applied the parallelogram CM, equal
to the parallelogram EH, and fo pofited, as that AC and
CN may be direct: Therefore (a) DB and BM fhall be
alſo direct (that is, as one right-line.)
Demonftr. Seeing that the ratio of AB to EH is gi-
ven, and that CM is equal to EH, the ratio of AB to
CM is alſo given; and therefore the ratio of AC to CN
is given (feeing A B is to CM, (b) as AC is to CN ;)
and as CM is equal and equiangled to EH, the
fides about the equal angles of the parallelograms CM and
EH. (c) are reciprocally proportional; and therefore as
CB is to FH, fo is EF to CN. But the ratio of AC
to CN is given: Therefore as CB is to FH, ſo is EF to
that to which AC hath a given ratio.
Conftr. 2. Now fuppofe AB not to be equiangled to
EH, and in the given point C of the line CB, let there
be conftituted the angle BCK equal to the angle EFH,
and finiſh the parallelogram CL.
Demonftr. 2. Seeing then that the ratio of AB to EH
is given, and (d) that AB is equal to CL, alfo the ratio
of CL to EH is given, and the angle BCK is equal
to the angle F, and therefore CL (e) is equiangled
to EH: Therefore (by the first part of this propofi-
tion) as CB is to FH, fo is EF to that to the which CK
hath a given ratio. But the ratio of AC to CK
is given; (as appears by what hath been demonſtra-
ted in the latter part of the precedent propoſition.)
Therefore as CB is to FH, fo is EF to that to
which AC hath a given ratio,
PROP.
EUCLIDE'S DATA.
309
PRO P. LXXV. Plate VIII. Fig. 40, 41.
If two triangles ABC and DEF, in equal angles A
and D, or elle unequal (yet nevertheless given) have to
one another a given ratio, as the fide AB of the first, fhall
be to the fide DE of the ſecond, ſo the other fide DF of the
Second, fhall be to that right-line to the which the other
fide AC of the first hath a given ratio.
Conftr. For let the parallelograms AG and DH be
finiſhed.
Demonfir. Forafmuch as the ratio of the triangle
ABC to the triangle DEF is given, alfo the ratio of
the parallelogram AG to the parallelogram DH is gi
ven.
Seeing therefore that the two parallelograms AG and
DH in equal angles, or unequal angles (nevertheleſs gi-
ven) have to one another a given ratio; as (a) AB is a 74. prop.
to DE, fo is DF to that to which AC hath a gi-
ven ratio.
PRO P. LXXVI. Fig. 4.
If from the top A of a triangle ABC, given in kind,
there be drawn to the bafe BC, a perpendicular line AD,
that line AD fhall have to the bafe BC a given ratio.
Demonftr. For feeing that the triangle ABC is gi-
ven in kind, the ratio of AB to BC is given;
and the angle B is alfo given. But the angle ADB
is given; therefore the other angle BAD is given.
Wherefore (a) the triangle ADB is given in kind :
and therefore the ratio of A B to AD is given.
But the ratio of AB to BC is given: Therefore (b) b 8. prop.
the ratio of AD to BC is given.
PROP. LXXVII. Fig. 44, 45.
If two figures ABC and DEF, given in kind, have
to one another a given ratio, the ratio alſo ſhall be given
of which you please of the fides of one of the figures, to
which you please of the fides of the other figure.
U 3
a 40. prop.
Confir.
མཐཱཀ༔ ཀྭ
310
EUCLIDE'S DAT A.
a 49. prop.
b 8. prop.
Confir. For on the right-lines BC and EF, let there
be deſcribed the fquares BG and EH.
ven
Demonftr. Forafmuch as on one and the fame right-
line BC, are deſcribed two figures ABC and BG gi-
in kind, (a) the ratio of the faid ABC to BG
is given. In like manner the ratio of DEF to EH
is given; and feeing that the ratio of ABC to DEF
is given; and alſo that of the fame figure ABC to
BG; and again the ratio of DEF to EH: (b) the
ratio of BG to EH is given; and therefore the ratio
of BC to EF is alfo given.
PRO P. LXXVIII. Plate VIII. Fig. 46, 47-
If a given figure ABC, hath a given ratio to fome
rectangled figure DF, and that one fide BC hath a gi-
ven ratio to one fide DE, the rectangled figure DF is
given in kind.
Conftr. For on the right-line BC let the fquare BH
be defcribed, and to the right-line DE, let the pa-
rallelogram DK be applied equal to BH, in fuch a
manner, as that GD and DI may be placed directly,
a fch. 68.prop. (a) and by confequence FE and EK alfo directly.
b 49, prop.
c 8. prop.
d 14. 6.
Demonftr. Therefore feeing that on one and the fame
right-line BC are defcribed the two rectiline figures ABC
and BH, given in kind, (b) the ratio of ABC to BH is
given. But the ratio of the faid ABC to DF is alfo gi-
ven: Therefore (c) the ratio of BH to DF is given. But
BH is equal to DK: Therefore the ratio of DK to DF
is alfo given. And feeing that BH is equal and equian-
gled to DK, both the one and the other being rectangles,
(d) the fides of thoſe figures are reciprocally porportional;
and as BC is to DE, fo is DI to CH. But by fuppofi-
tion, the ratio of BC to DE is given; therefore alfo the
ratio of DI to CH is given; but the ratio of DI to DG
is alfo given: (for DI is to DG (e) as DK to DF
Therefore (f) the ratio of DG to CH is given. But
CH is equal to BC, feeing that BH is a fquare; there-
fore the ratio of the fame BC to DG is given: But the
ratio of the fame BC to DE is alfo given; therefore the
ratio of DE to DG is given, and the angle at D is a right
gfch. 61.prop angle: Therefore (g) DF is given in kind,
e 1.6,
£ 8. prop.
PROP.
EUCLIDE'S DATA.
311
PRO P. LXXIX. Plate VIII. Fig. 48, 49.
If two triangles ABC and EFG, have an angle B equal
to an angle F. And from the equal angles B and F there be
drawn perpendiculars BD and FH, to the bafes AC and
EG; and that as the baſe AC of the first triangle ABC, is
to the perpendicular BD, fo alfo the bafe EG of the other.
triangle EFG, is to the perpendicular FH, thofe triangles
ABC and EFG are equiangled.
Conftr. For about the triangle EFG let there be de-
fcribed the circle EFLG, then on the right-line EG,
and in the point E given therein, let there be made the
angle GEL, equal to the angle C, and let FL and LG
be drawn, and the perpendicular LM.
Demonftr. Seeing then that the angle GEL is equal
to the angle C, and the angle ELG is equal to the angle
EFG, (a) they being in one and the fame fegment of the
circle; the third angle EGL is equal to the third angle
A. Wherefore the triangle ABC is alike to the triangle
ELG, and the perpendiculars BD and LM are drawn:
Therefore † as AC is to BD, fo is EG to LM; but by
fuppofition as AC is to BD; fo is EG to FH: Therefore
(6) LM is equal to FH. But the faid LM is (c) parallel
to FH: Therefore (d) FL is alſo parallel to EG; and
therefore the angle FLE (e) is equal to the angle LEG.
But the angle C is alfo equal to the faid angle LEG, and
the angle FLE to the angle FGE (f): Therefore alſo the
angle C is equal to the angle FGE. But by fuppofition
the angle ABC is equal to the angle EFG: Therefore
the third angle BAC is equal to the third angle FEG:
Wherefore the triangle ABC is equiangled to the trian-
gle EFG.
Scholium.
+ Now that as AC is to BD, ſo EG is to LM, it is by
fome thus demonftrated. Forafmuch as the angle C is
equal to the angle GEL, and the angle BDC to the angle
LME, each being a right-angle, the other angle CBD is
equal to the other angle ELM: Therefore (g) as EM is to
ML, fo is CD to DB. Again, feeing the angle ABC is
equal to the angle ELG, and the angle CBD to the angle
ELM, the remaining angle ABD is equal to the remaining
angle MLG; but the angle ADB is alfo equal to the
angle LMG; and therefore the third angle A is equal to the
U 4
third
a 21. 3..
b7. 5.
C 28. t.
d 33. 1.
e 29. L.
£ 21. 3.
84.60
312
EUCLIDE'S DATA.
h 4.6.
i 14. 5.
a 40. prop.
b 1.6.
€ 41. I.
d 8. prop.
e 1. 6.
f 8. def.
8 4. def.
h 2. prop.
i 27. prop.
k 28. prop.
125. prop.
third angle LGM: Therefore (h) as AD is to DB, fo is
GM to ML. But it hath been demonftrated, that as CD
is to DB, fo is EM to ML: Therefore (i) as AC is to BD,
fo is EG to LM.
PRO P. LXXX. Plate VIII. Fig. 5c, 51.
If a triangle ABC hath one angle A given, and that the
rectangle contained under the fides AB and AC, compriſing
the given angle A, hath a given ratio to the Square of the
other fide BC, the triangle ABC is given in kind.
Conftr. For from the points A and B, let there be
drawn the perpendiculars AD and BE.
Demonfir. Forafmuch as the angle BAE is given, and
alfo the angle AEB, the triangle ABE is given in (a)
kind; and therefore the ratio of AB to BE is given:
Therefore the ratio of the rectangle of AB and AC to
the rectangle of BE and AC is alfo given (for it is the
fame ratio (b) as of AB to BE.) But the rectangle of
AC and BE is equal to the rectangle of BC and AD;
for that each of thofe rectangles is (c) double to the
triangle ABC. Therefore the ratio of the rectangle of AB
and AC to the rectangle of BC and AD is alto given.
But the ratio of the rectangle of AB and AC to the
fquare of BC is given: Therefore (d) alfo the ratio of
the rectangle of BC and AD to the fquare of EC is
given; and therefore the ratio of the right-line BC to
the right-line AD is given. (For that (e) the rectangle
is to the fquare as AD to BC.) Now let the right-line
FD, given in pofition and magnitude, be expoſed;
and thereon let there be defcribed the fegment of a
circle FID; capable of an angle equal to the angle A.
And feeing the faid angle A is given, alfo the angle in
the fegment FLD fhall be given; and therefore (f) the
fame fegment is given in pofition. From the point D
let there be erected at right-angles on the line FD, the
line DH, which (g) is given in pofition: Let it be fo
made, that as EC is to AD, fo FD may be to DH and
ſeeing that the ratio of BC to AD is given, alſo that of
FD to DH is given. But FD is given: Therefore (b)
DH is given in magnitude. But it is alfo given in po-
fition, and the point D is given: Therefore the point
His (1) alfo given. Now through the point H let
there be drawn HI, parallel to FD, and that line HI
ſhall be given in (k) pofition. But the fegment of the
circle FID is alfo given in pofition. Therefore (1) the
point
EUCLIDE'S DATA.
313
point I is given. Let the right-lines IF and ID be
drawn, and the perpendicular IE: Therefore IE is given
in pofition. But the point I is given, as alfo each of
the points F and D: Therefore (m) each of the lines
FD, FI, and ID is given in pofition and magnitude:
Wherefore (2) the triangle FID is given in kind; and
feeing that as BC is to AE, fo is FD to DH, and (0)
that to DH, IE is equal; as BC is to AE, fo is FD
to IE, and the angle A is equal to the angle FID :
Therefore (p) the triangle ABC is equiangled to the
triangle FID. But FID is given in kind: Therefore
alſo the triangle ABC is given in kind.
OTHER WISE. Fig. 52, 53.
Confir. Let the triangle ABC, whofe angle A is gi-
ven, and the ratio of the rectangle contained under AB
and AC, to the fquare of BC be given: I fay that
the triangle ABC is given in kind.
Demon/ir. For feeing the angle A is given, that ſpace
by which the fquare of the line compounded of BAC is
greater than the fquare of BC, (9) hath a given ratio
to the triangle ABC. Now let that ſpace be Ď: There-
fore the ratio of D to the triangle ABC is given. But
the ratio of the triangle ABC to the rectangle of AB and
AC is given; () ſeeing the angle A is given: There-
fores the ratio of the ſpace D to the rectangle of
AB and AC is given. But the ratio of the rectangle
of AB and AC to the fquare of BC is alfo given:
Therefore (s) the ratio of the ſpace D to the fquare of
BC is given. Wherefore by compounding, (t) the ratio
of the ſpace D, with the fquare of BC the faid fquare
of BC is given: Therefore the ratio of the ſquare of the
line compounded of BAC, to the fquare of BC is given;
(for that the ſpace D with the fquare of BC is equal
to the fquare of the line compounded of BAC;) and
therefore (u) the ratio of the faid line compounded of
BAC to BC is given. But the angle A is alfo given :
Therefore (x) the triangle ABC is given in kind.
PRQ P.
LXXXI.
If of three right-lines A, B, and C,
proportional to three other proportional
right-lines D, E, and F, the extremes
A and D, C and F, are in a given ra-
tio, (to wit, as A to D, and C to F,)
also the means, В and E shall be in a
given ratio, and if one extreme hath a
A
·|-101
| -| -| -
given
m 26. prop.
n 39. prop.
0 34. prop.
P 79. prop.
q 67. prop.
r 66. prop.
s 8. prop.
t 6. prop.
u ſch. 52.
prop.
x 46. prop.
314
EUCLIDE'S DATA.
1
a 70. prop.
b 17.6.
c ſch. 52.
prop.
d 50. prop.
68. prop.
given ratio to an extreme, and the mean to the mean, the
other will have alſo a given ratio to the other.
Demonftr. Forafmuch as the ratio of A to D, and of
C to F is given, the rectangle of A and D (a) fhall have
a given ratio to the rectangle of C and F. But the
rectangle of A and D is equal (b) to the fquare of B; and
the rectangle of C and F to the fquare of E. Therefore
the ratio of the fquare of B to the fquare of E is given;
and therefore (c) the ratio of the line B to the line E is
alfo given.
Again, let the ratio of A to D, and B to E, be given:
I fay that the ratio of C to F is alfo given. For feeing
that the ratio of A to D, and of B to E is given, alfo
the ratio of the fquare of B (d) to the fquare of E is gi-
ven. But the ſquare of B is equal to the rectangle of A
and C, and the fquare of E to the rectangle of D and F:
Therefore the ratio of the rectangle of A and C to the
rectangle of D and F is given. But the ratio of a fide
A to a fide D is given: Therefore (e) the ratio of the
other fide C to the other fide F is alſo given.
PROP. LXXXII.
XXXII.
If there be four right-lines A,B,C, and
D, proportional, as the first A shall be to
that line to which the fecond В hath a
given ratio, fo the third C shall be to
that to which the fourth D hath a gi-
ven ratio.
A
B
C
D
E
F
Conftr. Let E be the line to which B hath a given
ratio, and let it be fo as that B may be to E, as D
is to F.
Demonftr. Now the ratio of B to E is given, therefore
alfo the ratio of D to F is given. And feeing that as
A is to B, fo is C to D. And again, as B is to E, fo
is D to F, by ratio of equality, as A is to E, fo is C to
F. But E is that line to which B hath a given ratio,
and F that to which D alſo hath a given ratio: There-
fore as A is to that to which B hath a given ratio, ſo C
is to that to which D hath a given ratio.
PROP.
EUCLIDE'S DATA.
315
PROP. LXXXIII,
If four right-lines A, B, C, and D,
are in fuch fort to one another, that of
any three of them A, B, C, and a fourth
E, taken proportional, to which that line
D, which remains of the four lines, hath
a given ratio, it shall be as the fourth D is
to the third C, fo the ſecond В ſhall be to
that to which the first A hath a given
ratio.
A
B
C
D
E
a 16. 6.
Demonftr. Forafmuch as A is to B, as C is to E, the
rectangle contained under A and E (a) is equal to the
rectangle contained under B and C; and feeing that the
ratio of D to E is given, alfo fhall be given the ratio of
the rectangle of A and D to the rectangle of A and E
(for (b) it is the fame ratio as of D to E.) But the rectan-
gle of A and E is equal to the rectangle of B and C.
Therefore the ratio of the rectangle of A and D to the
rectangle of B and C is given. Wherefore (c) as D is to c 56. propa
C, fo is B to that to which A hath a given ratio.
PRO P. LXXXIV. Plate VIII. Fig. 54.
If two right-lines AB and AE comprehending a given
face AF in a given angle BAE, and that the one AB be
greater than the other AË by a given line CB, alfo each of
the lines AB and AE is given.
Demonftr. For feeing that AB is greater than AE by
the given line CB, the remainder AC is equal to AE:
Finish the parallelogram AD. Therefore feeing that
AE is equal to AC, the ratio of AE to AC is given,
and the angle A is alfo given: Therefore (a) AD is given
in kind. Wherefore the given ſpace AF is applied
to the given right-line CB, exceeding it by the given
figure AD given in kind; and therefore (b) the breadth
of the excess is given. Therefore AC is given. But
CB is alfo given: Therefore the whole AB is given.
But AE is alfo given: Therefore each of the right-
lines AB and AE is given.
b 1.6.
a fch. 61.
prop.
b 59. prope
PROP
316
EUICLDE'S DATA.
a fch. 61.
prop.
PROP. LXXXV. Plate VIII. Fig. 54.
If two right-lines AC and CD, comprehend a given
Space AD in a given angle ACD, the line compounded of
thofe lines AC and CD is given, alſo each of thoſe lines
AC and CD is given.
Conftr. For let AC be prolonged to the point B, and
let CB be put equal to CD, then through the point
B let BF be drawn parallel to CD, and fo finiſh the
parallelogram CF.
:
Demonftr. Seeing then that CB is equal to CD, and
the angle DCB is given; for that angle that follows
is the given angle; and therefore (a) the parallelogram
DB is given in kind and again, feeing that the line
compounded of ACD is given, and CB is equal to
CD, alfo AB is given. And thus to the right-line AB
there is applied the given ſpace AD, deficient by the
b 58. prop. figure DB given in kind: and therefore (b) the breadths
of the defects are alfo given: Therefore the right-lines
DC and CB are given. But the compounded line
ACD is alſo given: Therefore (c) each of the lines AC
and CD is given.
€ 4. prop.
211. def.
b z. 2.
PRO P. LXXXVI.
Fig. 55.
If two right-lines AB and BC, comprehend a given
Space AC, in a given angle ABC, the fquare of the one
BC, is greater than the Square of the other AB, by
given Space (yet in a given ratio,) alfo each of those lines
AB and BC shall be given.
Demonftr. For feeing that the fquare of BC is greater
than the fquare of AB by a given ſpace (yet in a cer-
tain ratio) Let the given space be taken away, that
is to fay, the rectangle contained under CB and BE:
Therefore (a) the ratio of the remainder, (b) which is the
rectangle contained under BC and CE to the fquare
of AB is given. And forafmuch as the rectangle
† under AB and BC is given, and alſo that of CB and
BE, their (c) ratio is given. But as the rectangle under
AB and BG is to the rectangle under CB and EB, (d) fo
AB is to BE; and therefore the ratio of AB to BE is
e 50. prop. given: Wherefore (e) the ratio of the fquare of AB to
the ſquare of BE is alfo given. But the ratio of the
fquare of AB to the rectangle under BC and CE is gi-
ven: Therefore (ƒ) alſo the ratio of the rectangle under
c 1. prop.
d 1. 6.
£ 8. prop.
{
BC
EUCLIDE'S DATA.
317
h 8. z.
i 54. prop.
k 6. prop.
BC and CE to the fquare of BE is given. Wherefore
the ratio of four times the rectangle under BC and CE
to the fquare of BE is given; and by compounding, (g) g6. prop.
the ratio of four times the rectangle under BC and CE,
with the fquare of BE the fquare of BE is given
But four times the rectangle of BC and CE, with the
ſquare of BE, () is the fquare of the compound line
BCE: Therefore the ratio of the fquare of the compound
line BCE to the fquare of BE is given: Wherefore) the
ratio of the line compounded of BC and CE to BE is
given, and by compounding (k) the ratio of the com-
pound of the lines BC, CE, and BE, that is to ſay,
the double of BC to BE is given; and therefore the
ratio of the only line BC to BE is alſo given. But as
BC is to BE, () fo the rectangle under BC and BE is
to the fquare of BE: Therefore the ratio of the rectan-
gle under BC and BE to the fquare of BE is given.
But the rectangle of BC and BE is given: Therefore (m)
the fquare of BE is alfo given, and confequently the
line BE is given. Wherefore BC is alfo given, feeing
that the ratio of BE to BC is given. But the ſpace
AC is given, and alfo the angle B: Therefore (») ´AB____n 57. prop.
is given. Wherefore each of the lines AB and BC is
given.
Scholium.
+ Instead of faying in this place [what is under, &c.]
we have uſed this word rectangle, it being manifeft by
what follows that fuch was the intention of EUCLIDE,
Seeing that he makes ufe in the faid Demonftration of the fecond,
and eighth propofition of the Second Element; and also
that the space or parallelogram being given not rectangled,
it may be reduced thereto, making on BC, and in the gi-
ven point В, a right-angle CBA, ſo as that there will be
two parallelograms conftitued on one and the fame baſe
BC, and between the fame parallels, as in the 69th propo-
fition, by means whereof this conclufion is drawn.
Note, This ferves alfo for the next Prop.
PRO P. LXXXVII. Fig. 56.
If two right-lines AB and BC, comprehend a gi-
ven Space AC, in a given angle В, the Square of the
one BC is greater than the fquare of the other AB, by
a given space; alſo each of those lines AB and BC shall
be given.
Demonfir.
11.6,
m 2. prop.
318
EUCLIDE'S DATA.
a 2. 2.
b 1.6.
• 50. prop.
d 6. prop.
e 8. 2.
€ 54. prop.
g 6. prop.
h 8. prop.
i 1.6.
kz. prop.
Demonßr. For feeing that the fquare of BC is greater
than the fquare of AB by a given fpace: Let the given.
ſpace be taken away, and let the rectangle be contained
under BC and BE: Therefore the remainder (a) which
is the rectangle of BC and CE, is equal to the fquare of
AB. And feeing that the rectangle of BC and BE is
given, and alfo the ſpace or rectangle AC, the ratio of
the faid rectangle of BC and BE to AC is given. But
as (6) the rectangle of BC and BE is to the rectangle
of AB and BC, fo is BE to AB: Therefore the ratio of
BE to AB is given, and therefore (c) the ratio of the
fquare of the faid DE to the fquare of AB is alfo given.
But to that fquare of AB the rectangle of BC and CE
is equal: Therefore the ratio of the faid rectangle of
BC and CE to the fquare of BE is given; and therefore
the ratio of the quadruple of the faid rectangle of BC
and CE to the fquare of BE is alfo given; and by
compounding (d) the ratio of four times the rectangle
of BC and CE, with the fquare of BE to the faid
fquare of BE is given. But four times the rectangle of
BC and CE, with the fquare of BE, (e) is the fquare of
the compound line BCE: Therefore the ratio of the
fquare of that compound line BCE to the fquare of BE is
alfo given; and therefore the ratio (f) of the compound
line BCE to BE is given. Wherefore by compounding
g) the ratio of the faid compound line BCE and EB,
that is to fay, twice BC to BE is alfo given; therefore
the ratio of the only line BC to BE is given. But the
ratio of the fame BE to AB is alfo given: Therefore
(b) the ratio of AB to BC is given. And feeing that the
ratio of BC to BE is given, and that as the faid BC is
to BE, fo is the fquare of BC (i) to the rectangle of BC
and BE, the ratio of the fquare of BC to the rectangle
of BC and BE is alfo given. But the ſaid rectangle of
BC and BE is given, it being that which was taken
away, and which was given. Therefore the fquare of
BC (k) is given, and therefore the line BC is given.
But the ratio of the fame BC to BA is given, therefore
AB is alfo given.
PRO P. LXXXVIII. Plate VIII. Fig. 57.
If in a circle ABC, given in magnitude, there be drawn
e right-line AC, which ſhall take away a fegment ABC,
which doth comprehend a given angle AEC, that line AC
is given in magnitude.
Conftr.
EUCLIDE'S DATA.
319
Conftr. For let D be the center of the circle; and let
the diameter thereof ADE be drawn, and let EC be
joined.
Demonftr. The angle ACE is given for (a) it is a right-
angle. But the angle AEC is alfo given, and there-
fore the other angle CAE is given. Wherefore the tri-
angle ACE (b) is given in kind; and therefore the
ratio of EA to AC is given. But AE is given in
magnitude, ſeeing that the circle ABC is given in mag-
nitude. Therefore (c) AC is alfo given in magnitude.
PRO P. LXXXIX. Plate VIII. Fig 57.
If in a circle ABC, given in magnitude, there be drawn
a right-line AC, given in magnitude, that line AC will
take away a jegment ABC, comprehending a given angle.
Conftr. For having taken the point D for the center
of the circle, let the diameter ADE be drawn, as alſo
the right-line EC.
Demonftr. Forafmuch as each of the right-lines AE
and AC are given, the ratio of the line AE to AC (a) is
given; and the angle ACE is a right-angle: Therefore
(b) the triangle AČE is given in kind, and therefore
the angle AEC is given.
PROP. XC. Fig. 58.
If in the circumference of a circle ABC, given in pofition
and in magnitude, there be taken a given point B, and that
from the point B to the circumference of the circle ABC, a
right-line BAC be inflected fo as to make a given angle BAC,
the other extremity C of the inflected-line fhall be given.
Conftr. For let the center of the circle be D, and let
the right-lines BD and BC be drawn.
Demonftr. Forafmuch as each point B and D is given,
the right-line BD, (a) is given in pofition; and feeing
that the angle BAC is given, the angle BDC is alfo
given. Wherefore to the right-line BD, given in pofi-
tion, and in the point D given therein, there is drawn
the right-line CD; which makes the given angle BDC;
and therefore (b) the line DC is given in pofition.
But the circle ABC is given in pofition and magnitude:
Therefore (c) the right-line DC is given in pofition and
in magnitude. But the point is D given: Therefore (d)
the point C is alfo given.
PROP.
a 31. 3.
b 40. prop.
c 2. prop.
a 1. prop.
a 1.
b 43. prop.
a 26. prop.
b 29. prop.
c 6. def.
d 27. propò
320
EUCLIDE'S DAT A
a 26. prop.
b 18. 3.
c 6 def.
d 25. prop.
e 26. prop.
a 91. prop.
b 52. prop.
€ 36. 3.
d 26. prop.
e 25. prop.
PROP. XCI. Plate VIII. Fig. 59.
If from a given point C, there he drawn a right-line
CA, which shall touch a circle AB, given in pofition; that
line CA is given in pofition and in magnitude.
Conftr. For having taken the point D for the center
of the circle, let the right-lines DA and DC be drawn.
Demonftr. Forafmuch as each point C and D is given,
the right-line CD (a) is given in pofition and in mag-
nitude. But the angle CAD (b) is a right-angle; and
therefore the femicircle defcribed on CD fhall pafs
through the point A Let it then país through that
point, and let the femicircle be DAC: Forafmuch as
the fame DAC (c) is given in pofition, and alfo the
circle ABE, (d) the point A is given. But the point.
C is alfo given: Therefore (e) the right-line AC is
given in pofition and in magnitude.
PRO P. XCII. Fig. 60.
If without a circle ABC, given in pofition, there be taken
Some point D, and from that given point there be drawn a
right-line DB, cutting the circle, the rectangle comprifed
under the whole line BD, and the part DC, between the
point D, and the convexity of the circumference AC, ſhall
be given.
Conftr. For from the point D let the right-line DA
be drawn, which fhall touch the circle in the point A.
Demonftr. Therefore DA (a) is given in pofition and
magnitude; and therefore the fquare of the faid DA is
(b) given. But the faid fquare of DA is equal (c) to the
rectangle of BD and DC: Therefore the faid rectangle
of BD and DC is alfo given.
OTHERWISE. Fig. 61.
Conftr. Let E be the center of the circle, and through
the fame center let there be drawn from the point D
the right-line DA.
Demonftr. Forafmuch as each point D and E is given,
the right-line DE is (d) given in pofition and in magni-
tude. But the circle ABC is given in pofition and in
magnitude: Therefore each point A and F (e) is given,
and the point D is alfo given; and therefore each line
AD
Plate VIII. Facing Pag. 320.
A
Figs.
Fig. 3.
A
Fig.4.
Fig.5.
A
Fig. 6.
Fig.2.
B
A
Fig.7.
B K
H
D
F
D
K
E
BED
E
F
G
୯
E
B
E
B
Fig.8.
Fig.9.
Fig.
20.
G
E
B D
C
E
A
B
F
Fig.1.
Fig.12.
E
D
с в
H
D
D
H
F
B
Fig 17.
D
A
Fig17
Fig18.
H
A
C
H
21.
E C
Fig. 13. Fig14. Fig.15.
F Fig. D
10
A
.B
B
A D
Figz6.
Le Figag.
K
E
D
C
Fig.20.
A
D
B
H
H
E
E
N
F
M
D
I
K HI
K
M N
B
H
G
0
B
B
A
Fig.22
Fig.23.
H
K
H
M
F
G
K
D
E
I
B
B
E
Fig.24.
P
F
E
A
G
E
Fig 25.
E
ID
H
C
A
Fig.ro
G
F D
I
D
A
C
B
E
L
F
F
Fig.28
A
DD
Fig.29.
E
I
Ak
C
୯
B
Fig.27
B
A
D
Fig.40.
F
Fig.
32.
B
E
Fig, 30.
Fig.31.
A
KD
F
B
B
E
B F H C
K
K G
H
A KE
Fig.33.
E
E
B
Fig.
Fig.
34.
35.
Fig.36
Fig.37
Fig. 38.
A
D
F
B
H
B F
Η
A K
D
K
DE
D
F KML
୯
Fig.41.
A
E
F
Efig. 4
Fig.43.
B
CE
Fig.39.
Fig.44
Fig.45
M
F
B
CE
H
G
H
I
B
B
Fig.46.
Fig. 52.
A
Fig.48.
Fig49.
E
D
Fig.50⋅ Fig
Fig.51.
Fig.47.
B
F
D
E
H
G
M
B
E
T
Fig.53.
D
K
E
D F
D
F
Λ
D
A
Fig.57
B
Fig.54.
C B
Fig. 60.
Fig 55. Fig.56.
A
E
D
AFig. 58.
B
B
୯
E
E B
D
B
B
c
D
B
Fig.62.
Fig.59.
E
B
A
୯
F
EUCLIDE'S DATÀ.
321
AD and FD is given. Wherefore the rectangle of the
lines AD and DF is alfo given. But the faid rectangle
of AD and DF is equal to the rectangle of DB and DC:
Therefore the rectangle of DB and DC is given.
PRO P. XCIII. Plate IX. Fig. 1.
If in a circle given in pofition there be taken a given point
A, and through that point A there be drawn a right-line BC
to [the circle, the rectangle compriſed under the fegments
the fame line BC, fhall be given.
Conftr. For let D be taken for the center of the circle,
and having drawn the right-line AD prolong it to the
point E and F.
Demonftr. Forafmuch as each point A and D is given,
the right-line AD (a) is given in pofition. But the circle
BEC is alſo given in poſition: Therefore each point E
and F is alfo given in pofition, and the point A is gi-
ven. Wherefore each line (b) AE and AF is given:
Therefore the rectangle of the fame lines AE and AF is
given, and is equal to the rectangle (b) of AB and AC:
Therefore the faid rectangle of AB and AC is given.
PRO P. XCIV. Fig. 2.
If in a circle ABC, given in magnitude, there be drawn
a right-line BC, which takes away a segment which
comprehends a given angle BAC, and that the faid
angle in the fegment is cut into two equal parts, the line
compounded of the right-lines BA and AC, which comprehend
the given angle BAC ſhall have a given ratio to the line
AD, which divides that angle into two equal parts;
and the rectangle contained under the line compounded of
thofe lines BA and AC, comprehending the given angle
BAC, and that part ED of the interfecting-line which is
below the fegment between the bafe BC and the circumfe-
rence, shall be given.
Conftr. Let BD be drawn.
Demonftr. Foraſmuch as in the circle ABC given in
magnitude, there is drawn the right-line BC, which
takes away the fegment BAC, comprehending the given
angle BAC, that line BC (a) is given; and therefore BD
is alfo given: Therefore the ratio of BC to BD (b) is
given. And feeing that the given angle BAC is cut
in two equal parts by the right-line AD, as (c) BA
is to CA, fo is BE to CE; and by compounding, as
á 26. prop.
b 35• 3+
à 88. prop.
b 1. prop.
© 3.6.
X
BAC
322
EUCLIDE'S DATA.
d 21. 3.
e 4. 6.
f 15. I.
g 16. 6.
h 32. 1.
is I
5 ·
1
k 21.3.
3
BAC is to CA, fo is BC to CE; and by permutation,
as BAC is to BC, fo is CA to CE. And feeing that
the angle BAE is equal to the angle CAE, and the
angle ACE (d) to the angle BDE, the other angle AEC
is equal to the other angle ABD; and therefore the
triangle ACE is equiangled to the triangle ABD :
Therefore (e) as AC is to CE, fo is AD to BD. But
as CA is to CE, fo the line compounded of BA and
AC is to BC. Therefore as the compound-line BAC
is to BC, fo is AD to BD; and by permutation, as the
compound-line BAC is to AD, fo is BC to BD. But
the ratio of BC to BD is given: Therefore the ratio of
the compound-line BAC to AD is alfo given. More-
over, I fay that the rectangle under the compound-line
BAC and ED is given. For feeing that the triangle
AEC is equiangled to the triangle BDE, (for the angle
ACE() is equal to the angle BDE, and the angle AĒC
(f) to the angle BED) as BD is to DE, fo is AČ to CE.
But as AC is to CE, fo is alfo the compound-line BAC
to BC: Therefore as the compound-line BAC is to BC,
fo is BD to DE. Wherefore the rectangle of the com-
pound-line BAC and DE (g) is equal to the rectangle
of BC and BD. But the rectangle of BC and BD is
given, (for that thofe lines BC and BD are given :)
Therefore the rectangle under the compound-line BAC
and ED is alfo given.
OTHERWISE. Plate IX. Fig. 4•
Confir. Let CA be prolonged to the point E, and let
AE be put equal to BA, and let BE and BD be joined.
Demonfir. Forafmuch as the angle BAC is double to
each of the angles CAD and AEB (for the angle BAC
is cut into two equal parts by the line AD, and equal
(b) to the two angles ABE and AEB, which (¿) `are
equal) the angle ABE is equal to the angle CAD, that
is to fay, (k) to the angle CBD; adding therefore the
common angle ABC, the whole angle ABD fhall be
equal to the whole angle FBE. But the angle ACB
is (A) equal to the angle ABD: Therefore the third
angle AFB is equal to the third angle BAD; and
therefore the triangle CEB is equiangled to the trian-
gle ABD Wherefore as CE is to CB, fo is AD to
BD. But the right-line CE is compounded of the two
lines CA and AB: Therefore as the compound-line
BAC is to CB, fo is AD to BD; and by permutation,
as
F
EUCLIDE'S DATA.
as the compound-line BAC is to AD, fo is CB to BD.
But the ratio of CB to BD is given, ſeeing that each of
thofe lines is given: Therefore the ratio of the com-
pound-line BAC to AD is alfo given. And feeing that
the triangle CEB is equiangled to the triangle FBD
(for the angle AFC is equal () to the angle BFD,
the angle ECB (m) to the angle ADB) as EC is to
CB, fo is BD to DF. But EC is equal to the com-
pound-line BAC: Therefore as the compound-line
BAC is to CB, fo is BD to DF. Wherefore (2)
the rectangle of the compound-line BAC and DF is
equal to the rectangle of CB and BD. But the rectan-
gle of CB and DC is given, fince each of the lines
CB and BD is given; Therefore the rectangle of the
compound-line BAC and DF is given.
OTHERWISE. Plate IX. Fig. 5.
Conftr. Let AC be prolonged to F, and let CF be put
equal to AB, and let the right-lines BD and DF be
drawn.
121.3.
323
m 16. 6.
n 16. 6.
0 26,29.3.
P 22. 3.
Demonftr. Forafmuch as BA is equal to CF, and (0) BD
to DC, the two fides AB and BD are equal to the two
fides CD and DF,.each to his correfponding fide, and the
angle ABD is equal to the angle DCF, (p)feeing that the
four fided figure ABDC is within the circle: Therefore
the baſe AD is (q) equal to the baſe DF, and the angle q 4. 1.
DAB to the angle DFC. But the angle BAD is given,
being the half of the given angle BAC, therefore the
angle DFC is ſo alfo. But DAF is alfo given: There-
fore the triangle ADF is given in kind. Wherefore
the ratio of FĂ to AD is given. But AF is the com-
pound of BA and AC, for that CF is equal to AB:
Therefore the ratio of the compound-line BAC to AD
is given: The fame demonftration will ferve to fhew
that the rectangle contained under the compound-line
BAC and ED is given alſo.
PROP. XCV. Fig. 6.
If in the diameter BC of a circle ABC given in poſi-
tion, there be taken a given point D, and from that point
D there be drawn a right-line DA, to the circumference of
the circle. And if from the fection of the faid line there
be drawn a right-line AE, perpendicular thereto, and
through the point E where that perpendicular meets with
X 2
the
X
324
EUCLIDE'S DATA.
a 25. prop.
b 26. fup.
CIS. I.
d 26. I.
e 27. prop.
€ 93. trop.
the circumference, there be drawn a parallel EF, to the firft
line drawn AD, that point F in which the parallel meets
with the diameter, is given; and the rectangle contained
under the parallel-lines AD and EF is alfo given.
Confir. Let the right-line EF be prolonged to the point
G, and let the right-line AG be drawn.
Demonfir. Forafmuch as the angle AEG is a right-
angle, the right-line AG is the diameter of the circle.
But BC is alio the diameter: Therefore the point H is
the center of the circle. Now the point D is given;
and therefore (a) the line DH is given in magnitude. But
feeing that AD is parallel to EG, and AH equal to GH;
(b) DH is equal to FH, and AD to FG: (for the angles
AHD and FHG (c) are equal, and DAH and FGH (dare
alfo equal.) But the line DH is given: Therefore FH is
alfo given. But each of thoſe lines DH and HF is alfo
given in pofition, and the point H is given: Therefore
the point F is alfo given. And feeing that in the
circle ABC given in pofition, is taken the given
point F, and through the fame is drawn the right-line
EFG; the rectangle under EF and FG (f) is given.
But FG is equal to AD. Therefore the rectangle com-
prehended under AD and EF is given. Which was to
be demonftrated.
The End of EUCLIDE'S DATA.
(325)
܀
***
******
A
BRIEF TREATISE
(Added by FLUSSAS)
O F
Regular Solids:
R
Egular Solids are faid to be compofed and mix'd
when each of them is transformed into other
Solids, keeping ftill the form, number and
inclination of the bafes, which they before had to
one another; fome of which yet are transformed into
mix'd Solids, and others into fimple. Into mix'd, as a
Dodecaedron and an Icofaedron, which are transform-
ed or altered, if you divide their fides into two equal
parts, and take away the folid angles fubtended by
plane fuperficial figures, made by the lines coupling thofe
middle fections; for the Solid remaining after the tak
ing away of thofe folid angles, is called an Icofidode-
caedron. If you divide the fides of a Cube and of an
Octoedrom
X 3
I
326
A TREATISE of
a 6. 4.
b 15. 13.
Octoedron into two equal parts, and couple the fections,
the folid angles fubtended by the plane fuperficies made
by the coupling-lines, being taken away, there fhall be
left a Solid, which is called an Exoctoedron. So that
both of a Dodecaedron, and alſo of an Icofaedron, the
Solid which is made fhall be called an Icofidodecaedron;
and likewife the Solid made of a Cube, and alſo of an
Octoedron, ſhall be called an Exoctoedron. But the
other Solid, to wit, a Pyramis or Tetraedron, is trans-
formed into a fimple Solid, for if you divide into two
equal parts each of the fides of the Pyramis, triangles
deſcribed of the lines which couple the fections, and
fubtending and taking away the folid angles of the
Pyramis, are equal and like unto the equilateral trian-
gles left in each of the bafes, of all which triangles
is produced an Octoedron, viz. a ſimple, and not a
compoſed Solid. For the Octoedron hath four baſes,
like in number, form, and mutual inclination with the
baſes of the Pyramis, and hath the other four bafes
with like fituation oppofite and parallel to the former.
Wherefore the application of the Pyramis taken twice,
maketh a fimple Octoedron, as the other Solids make
a mix'd compound Solid.
DEFINITION S.
I.
An Exoctoedron is a folid figure contained under fix equal
Squares, and eight equilateral and equal triangles.
II.
An Icofidodecaedron is a folid figure contained under twelve
equilateral, equal, and equiangled Pentagons, and twenty
equal and equilateral triangles.
PROBLEM I. Plate IX. Fig. 7.
To defcribe an equilateral and equiangled Exoctoedron,
and to contain it in a given sphere, and to prove that the
diameter of the fphere is double to the fide of the faid Exocto -
edron.
Conftr. Suppoſe a Sphere whofe diameter let be AB,
and about the diameter AB let there be deſcribed a ſquare
(a) and upon the fquare let there be deſcribed a Cube (b)
which let be CDEFQTVR; and let the diameter thereof
,、、་
be
REGULAR SOLIDS
327
be QR, and the center S. Divide the fides of the Cube
into two equal parts in the points G, H, I, K, L, M, N,
O, P, &c. and join the middle fections by the right-
lines IN, NO, OP, PI, and fuch like, which fubtend
the angles of the fquares or bafes of the Cube; and they
are equal (c) and contain right-angles, as the angle NIP.
For the angle NID, which is at the bafe of the lfofceles
triangle NDI, is the half of a right-angle, and fo like-
wife is the oppofite angle RIP. Wherefore the refidue
NIP is a right-angle, and fo the reft. Wherefore NIPO
is a fquare. And for the fame reaſon ſhall the reſt NMLK,
KGHI, &c. infcribed in the bafes of the Cube, be
fquares, and they fhall be fix in number, according to the
number of the bafes of the Cube. Again, forafmuch as
the triangle RIN fubtendeth the folid angle D of the
Cube, and likewife the triangle KGL the folid angle.
C, and fo the reft which fubtend the right folid angles
of the Cube, and theſe triangles are equal and equilateral
(to wit) being made of equal fides, and they are the li-
mits or borders of the fquares, and the fquares the limits or
borders of them; as hath been before proved. Where-
fore LMNOPHGK is an Exoctoedron by the definition,
and is equilateral, for it is contained by equal fubtendant
lines; it is alfo equiangled, for every folid angle thereof
is contained under two fuperficial angles of two fquares,
and two fuperficial angles of two equilateral triangles.
C 4. I.
Demonftr. Forafmuch as the oppofite fides and dia-
meters of the baſes of the Cube are parallels, the plane
extended by the right-lines QT and VR, fhall be a
parallelogram. And becauſe alfo in that plane lieth QR,
the diameter of the Cube, and in the fame plane alſo
is the line MH, which divideth the faid plane into two
equal parts, and alfo coupleth the oppofite angles of the
Exoctoedron: this line MH therefore divideth the diame-
ter into two equal parts (d); and alſo divideth it felf d cor.34.3.
in the fame point, which let be S, into two equal parts (e).
And by the fame reaſon may we prove that the rest of the
lines which join the oppofite angles of the Exoctoedron,
do in S, the center of the Cube, divide one another into
two equal parts, for each of the angles of the Exoctoe-
dron are fet in each of the bafes of the Cube. Wherefore
making the center the point S, with the diſtance SH of
SM defcribe a Sphere, and it ſhall touch every one of
the angles equidiftant from the point S.
X 4
And
e 4. 1.
¿
328
A TREATISE of
f 33. 3.
g 2.6.
a 30.
6.
b 15. 13.
€ 17. 13:
d 4. 1.
And becauſe AB the diameter of the fphere given,
is put equal to the diameter of the baſe of the cube, to
wit, to the line RT, and the fame line RT is equal
to the line MH (ƒ) which line MH coupling the oppofite
angles of the Exoctoedron, is drawn by the center.
Wherefore it is the diameter of the Sphere given which
containeth the Exoctoedron.
Laftly, forafmuch as in the triangle RFT, the line PO
cuts the fides into two equal parts, it fhall cut them
proportionally with the bafes, to wit, as FR is to
FP, fo fhall RT be to PO. (g) But FR is double to
FP by fuppofition: Wherefore RT, or the diameter
HM, is alfo double to the line PO, the fide of the Ex-
octoedron. Wherefore we have defcribed, &r. Which
was required to be done.
PROBLEM II. Plate IX. Fig. 8.
To defcribe an equilateral and equiangled Icofidodecaedron,
and to comprehend it in a ſphere given, and to prove that the
diameter being divided in extream and mean proportion,
maketh the greater fegment double to the fide of the Icofi-
dodecaedron.
Confir. Suppofe that the diameter of the fphere given
be NL, (a) divide the line NL, in extream and mean
proportion in the point I, and the greater fegment there-
of let be NI, and upon the line NI deſcribe a Cube; (b)
and about this Cube let there be circumfcribed a Dode-
caedron; (c) and let the fame be ABCDEFHKMO, and
divide each of the fides into two equal parts in the points
Q, R, S, T, V, X, Y, Z, P, ‹, &, G, &c. and join
the fections with right-lines, which fhall fubtend the
angles of the Pentagons, as the lines FG, GV, VQ, QY,
YR, RQ, VT, TX, XV, and ſo the reſt.
-
Demonfir. Forafmuch as thefe lines fubtend equal an-
gles of the Pentagons, and thofe equal angles are con-
tained by equal fides, to wit, by the half of the fides
of the Pentagons; therefore thofe fubtending lines are
equal. (d) Wherefore the triangles GQV, YQR, and
VXT, and the reft, which take away folid angles of
the Dodecaedron, are equilateral.
Again, forafmuch as in every Pentagon are defcribed
five equal right-lines, coupling the middle fections of the
fides, as are the lines QV, VT, TS, SR, and RQ,
they defcribe a Pentagon in the plane of the Pentagon of
the Dodecaedron. And the faid Pentagon is contained
in
די
REGULAR SOLIDS.
329
in a circle, to wit, whofe center is the center of a Penta-
gon of the Dodecaedron. For the lines drawn from that
center to the angles of this Pentagon are equal, becauſe
they are perpendiculars upon the bafes cut. (e) Wherefore
the Pentagon QRSTV, is equiangled. (f) And by the
fame reafon may the rest of the Pentagons defcribed in the
bafes of the Dodecaedron, be proved equal and like.
Wherefore thoſe Pentagons are twelve in number:
And forafmuch as the equal and like triangles fub-
tend and take away twenty folid angles of the Dodecae-
dron; therefore the faid triangles fhall be twenty in
number. Wherefore we have defcribed an Icofidodeca-
edron by the definition, which Icofidodecaedron is equi-
lateral; for that all the fides of the triangles are equal
and common with the Pentagons; and it is alfo equian-
gled: For each of the ſolid angles is made of two fuper-
ficial angles of an equilateral Pentagon, and of two fu-
perficial angles of an equilateral triangle.
Now let us prove that it is contained in the given
ſphere whoſe diameter is NL. Forafmuch as perpendi-
culars drawn from the centers of the Dodecaedron, to
the middle ſections of its fides, are the halfs of the lines
which couple the oppofite middle fections of the fides of
the Dodecaedron, (g) which lines alfo (b) do in the center
divide one another into two equal parts. Therefore
right-lines drawn from that point to the angles of the
Icofidodecaedron (which are ſet in thoſe middle fections)
are equal; which lines are thirty in number, according
to the number of the fides of the Dodecaedron, for each
of the angles of the Icofidodecaedron are ſet in the mid-
dle fections of each of the fides of the Dodecaedron.
Wherefore making the center of the Dodecaedron, and
the ſpace, any one of the lines drawn from the center to
the middle fections, defcribe a fphere, and it fhall pafs
through all the angles of the Icofidodecaedron, and
fhall contain it.
And forafmuch as the diameter of this folid, is that
right-line whofe greater fegment is the fide of the Cube
infcribed in the Dodecaedron, (i) which fide is NI by
fuppofition. Wherefore that folid is contained in the
fphere given, whofe diameter is put to be the line NL.
€ 12. 4.
£ 11. 4.
8 3. cor. of
17. 13.
h idem.
i 4. cor.
17.13•
Now let us prove that the great fegment of the diame-
ter is double to QV the fide of the folid. Forafmuch as
the fides of the triangle AEB are in the points Q and V
divided into two equal parts, the lines QV and BE are
parallels, (k) Wherefore as AE is to AV, fo is EB to VQ. kcor.39.1.
(4) But
330
A TREATISE of
I z. 6.
m 4. 6.
2. cor. of
17. 13.
(But the line AE is double to the line AV. Where-
fore the line BE is double to the line QV. (m) Now
the line BE is equal to NI, or to the fide of the Cube ;
(2) which line NI is the greater fegment of the diameter
NL. Wherefore the greater fegment of the diameter
given is double to the fide of the Icofidodecaedron in-
fcribed in the given fphere. Wherefore we have de-
fcribed, &c. Which was required to be done.
ADVERTISEMENT.
And
To the underſtanding of the nature of this Icofidode-
caedron, you muſt well conceive the paffions and pro-
prieties of both thefe folids, of whoſe baſes it confifteth,
to wit, of the Icofaedron and of the Dodecaedron.
altho' in it the bafes are placed oppofitely, yet have they
to one another one and the fame inclination. By reaſon
whereof there lie hidden in it the actions and paffions
of the other regular Solids. And I would have thought
it not impertinent to the purpoſe to have given the in-
fcriptions and circumfcriptions of this Solid, if want of
time had not hindered. But that the Reader may the
better attain to the underſtanding thereof, I have here
following briefly fhewn, how it may in or about every
one of the five regular Solids be infcribed or circumfcri-
bed; by the help whereof he may, with ſmall labour, or
rather none at all, having well poifed and confidered the
Demonſtrations appertaining to the forefaid five regular
Solids, demonftrate both the infcription of the faid Solids
in it, and the Infcription of it in the faid Solids.
Of the Infcriptions and Circumfcriptions of an Icofidodeca-
edron.
An Icofidodecaedron may contain the other five regu-
lar bodies. For it will receive the angles of a Dodecae-
dron in the centers of the triangles which fubtend the
folid angles of the Dodecaedron, which folid angles are
twenty in number, and are placed in the fame order in
which the folid angles of the Dodecaedron, taken away,
or fubtended by them, are. And for that reaſon it fhall
receive a Cube and a Pyramis contained in the Dodeca-
edron, when as the angles of the one are fet in the an-
gles of the other.
An Icofidodecaedron receiveth an Octoedron, in the
angles cutting the fix oppofite ſections of the Dodecae-
dron, even as if it were a fimple Dodecaedron.
And
REGULAR SOLIDS.
331
And it containeth an Icofaedron, placing the twelve
angles of the Icofaedron in the fame centers of the twelve
Pentagons.
It may alſo by the fame reafon be infcribed in each of
the five regular bodies, to wit, in a Pyramis, if you place
four triangular bafes concentrical with four bafes of the
Pyramis, after the fame manner that you inſcribed an
Icofaedron in a Pyramis; fo likewife may it be inſcribed
in an Octoedron, if you make eight baſes thereof con-
centrical with the eight bafes of the Octoedron. It hall
alſo be inſcribed in a Cube, if you place the angles which
receive the Octoedron in it, in the centers of the bafes of
the Cube. Again, you ſhall infcribe it in an Icofaedron,
when the triangles formed of the Pentagon bafes, are
concentrical with the triangles which make a folid
angle of the Icofaedron.
Laftly, it fhall be inſcribed in a Dodecaedron, if you
place each of the angles thereof in the middle ſections of
the fides of the Dodecaedron, according to the order of
its conftruction.
The oppofite plain fuperficies alfo of this folid are
parallels. For the oppofite folid angles are fubtended by
parallel plain fuperficies, as well in the angles of the
Dodecaedron fubtended by triangles, as in the angles
of the Icofaedron fubtended by Pentagons, which thing
may eaſily be demonftrated. Moreover, in this ſolid are
infinite properties and paffions, fpringing from the folids
whereof it is compofed.
Wherefore it is manifeft, that a Dodecaedron and an
Icofaedron mixed, are transformed into one and the
ſelf ſame ſolid of an Icofidodecaedron. A Cube alſo and
an Octoedron are mixed and altered into another ſolid,
to wit, into one and the fame Exoctoedron. But a Py-
ramis is transformed into a fimple and perfect folid, to
wit, into an Octoedron.
If we will frame theſe two folids joined together into
one folid, this only muſt we obſerve.
In the Pentagon of a Dodecaedron inſcribe a like
Pentagon, and let its angle be ſet in the middle ſections
of the Pentagon circumfcribed, and then upon the faid
Pentagon infcribed, let there be fet a folid angle of an
Icofaedron, and fo obferve the fame order in each of the
bafes of the Dodecaedron, and the folid angles of the
Icofaedron, fet upon theſe Pentagons, fhall produce a
folid confifting of the whole Dodecaedron, and whole
Icofaedron. In like manner, if in every baſe of the Ico-
faedron
332
A TREATISE of
faedron, the fides being divided into two equal parts, be
infcribed an equilateral triangle, and upon each of thoſe
equilateral triangles be fet a folid angle of a Dodecae-
dron, there fhall be produced the fame folid confifting of
the whole Icofaedron, and of the whole Dodecaedron.
And after the fame order, if in the baſes of a Cube be
infcribed fquares fubtending the folid angles of an Octa-
edron, or if in the bafes of an Octoedron be infcribed
equilateral triangles fubtending the folid angles of a Cube,
there fhall be produced a folid confifting of either of the
whole folids, to wit, of the whole Cube, and of the
whole Octoedron.
But equilateral triangles infcribed in the bafes of a
Pyramis, having their angles fet in the middle ſections
of the fides of the Pyramis, and the folid angles of a
Pyramis, fet upon the faid equilateral triangles, there
fhall be produced a folid confifting of two equal and like
Pyramids.
And now if in thefe folids thus compofed, you take
away the folid angle, there fhall be reftored again the
first compofed folids, to wit, the folid angles taken
away from a Dodecaedron and an Icofaedron compoſed
into one. there fhall be left an Icofidodecaedron, the
folid angles taken away from a Cube and an Octoedron
compofed into one folid, there fhall be left an Exoftoe-
dron. Moreover, the folid angles taken away from two
Pyramids compofed into one folid, there fhall be left an
Octoedron.
Of the nastire of a trilateral and equilateral Pyramis.
1. A trilateral equilateral Pyramis is divided into two
equal parts, by three equal fquares, which in the center
of the Pyramis cut one another into two equal parts,
and perpendicularly, and whofe angles are fet in the
middle fections of the fides of the Pyramis.
2. From a Pyramis are taken away four Pyramids like
unto the whole, which utterly take away the fides
of the Pyramis, and that which is left is an Octoedron,
infcribed in the Pyramis, in which all the folids infcri-
bed in the Pyramis are contained.
3. A perpendicular drawn from the angle of the Pyra-
mis to the bafe, is double to the diameter of the Cube
infcribed in it.
4. And a right-line joining the middle fections of
the oppofite fides of the Pyramis is triple to the fide of
the fame Cube.
5. The
REGULAR SOLIDS.
333
5. The fide alfo of a Pyramis is triple to the diame-
ter of the baſe of the Cube.
6. Wherefore the fame fide of the Pyramis is in power
double to the right-line which joins the middle fection
of the oppofite fides.
7. And it is in power fefquialter to the perpendicular
which is drawn from the angle to the baſe.
8. Wherefore the perpendicular is in power fefquiter-
tia to the line which joins the middle fections of the
oppofite fides.
9. A Pyramis and an Octoedron infcribed in it, alfo
an Icofaedron infcribed in the fame Octoedron, contain
one and the ſame ſphere.
Of the nature of an Octoedron.
1. Four perpendiculars of an Octoedron, drawn in
four bafes thereof from two oppofite angles of the faid
Octoedron, and joined together by thofe four baſes,
defcribe a Rhombus, or Diamond figure; one of whoſe
diameters is in power double to the other diameter.
2. For it hath the fame proportion that the diameter of
the Octoedron hath to the fide of the Octoedron.
3. An Octoedron and an Icofaedron infcribed in it,
do contain one and the fame ſphere.
4. The diameter of the folid of the Octoedron is in
power fefquialter to the diameter of the circle which
containeth the baſe, and is in power duple fuperbiparti-
ens tertias (that is, as 8 to 3,) to the perpendicular or
fide of the forefaid Rhombus; and moreover is in length
triple to the line which joins the centers of the next
bafes.
5. The angle of the inclination of the baſes of the
Octoedron, doth, with the angle of the inclination of the
bafes of the Pyramis, make angles equal to two right-
angles.
Of the nature of a Cube.
1. The diameter of a Cube is in power fefquialter to
the diameter of his bafe.
2. And is in power triple to his fide.
3. And unto the line which joins the centers of the
next baſes, it is in power fextuple.
4. Again, the fide of the Cube, is to the fide of the
Icofaedron infcribed in it, as the whole is to the greater
fegment.
5. Unto
334
A TREAT IS E
5. Unto the fide of the Dodecaedron, it is as the
whole is to the leffer fegment.
6. Unto the fide of the Octoedron it is in power
duple.
7. Unto the fide of the Pyramis it is in power fub-
duple.
8. Again, the Cube is triple to the Pyramis, but to the
Cube the Dodecaedron is in a manner double. Where-
fore the fame Dodecaedron is in a manner fextuple to
the faid Pyramis.
Of the nature of the Icofaedron.
1. Five triangles of an Icofaedron, make a folid
angle, the baſes of which triangles make a Pentagon.
If therefore from the oppofite bafes of the Icofaedron
be taken the other Pentagon by them defcribed; theſe
Pentagons fhall in fuch fort cut the diameter of the
Icofaedron which joins the forefaid oppofite angles,
that that part which is contained between the planes of
thefe two Pentagons fhall be the greater fegment, and
the refidue which is drawn from the plane to the angle,
ſhall be the leffer fegment.
2. If the oppofite angles of two bafes joined together,
be coupled by a right-line, the greater fegment of that
right-line is the fide of the Icofaedron.
3. A line drawn from the center of the Icofaedron to
the angies, is in power quintuple to half that line which
is taken between the Pentagons, or of the of half that
line which is drawn from the center of the circle which
containeth the forefaid Pentagon, which two lines are
therefore equal.
4. The fide of the Icofaedron containeth in power
either of them, and alfo the leffer fegment, to wit,
the line which falleth from the folid angle to the
Pentagon.
5. The diameter of the Icofaedron containeth in power
the whole line, which coupleth the oppofite angles of the
bafes joined together, and the greater fegment thereof,
to wit, the fide of the Icofaedron.
6. The diameter alfo is in power quintuple to the
line which was taken between the Pentagons, or to the
line which is drawn from the center to the circumference
of the circle which containeth the Pentagon compofed of
the fides of the Icofaedron.
7. The
REGULAR SOLID S.
335
7. The dimetient containeth in power the right-line
which coupleth the centers of the oppofite bafes of the
Icofaedron, and the diameter of the circle which contain-
eth the baſe.
8. Again, the faid dimetient containeth in power the
diameter of the circle which containeth the Pentagon,
and alfo the line which is drawn from the center of the
fame circle to the circumference: that is, it is quintuple
to the line drawn from the center to the circumference.
9. The line which coupleth the centers of the oppofite
baſes containeth in power the line which coupleth the
centers of the next bafes, and alfo the reft of that line of
which the fide of the Cube infcribed in the Icofaedron.
is the greater fegment.
10. The line which coupleth the middle fections of the
oppofite fides, is triple to the fide of the Dodecaedron
infcribed in it,
11. Wherefore, if the fide of the Icofaedron and the
greater fegment thereof be made one line, the third part
of the whole is the fide of the Dodecaedron inſcribed in
the Icofaedron.
Of the Dodecaedron.
1. The diameter of a Dodecaedron containeth in power
the fide of the Dodecaedron, and alſo that right-line to
which the fide of the Dodecaedron is the leffer fegment,
and the fide of the Cube infcribed in it is the greater
fegment, which line is that which fubtendeth the angle
of the inclination of the bafes, contained under two
perpendiculars of the bafes of the Dodecaedron.
2. If there be taken two bafes of the Dodecaedron,
diſtant from one another by the length of one of the fides,
a right-line coupling their centers being divided in ex-
treme and mean proportion, maketh the greater fegment
the right-line which coupleth the centers of the next
bafes.
3. If by the centers of five bafes fet upon one baſe,
be drawn a plain ſuperficies, and by the centers of the
bafes which are fet upon the oppofite baſe, be drawn
alſo a plain fuperficies, and then be drawn a right-line,
coupling the centers of the oppofite baſes, that right-line
is fo cut, that each of his parts fet without the plain
fuperficies, is the greater fegment of that part which is
contained between the planes.
4. The
336
A TREATIS E, &c.
4. The fide of the Dodecaedron is the greater feg-
ment of the line which fubtendeth the angle of the
Pentagon.
5. A perpendicular-line drawn from the center of the
Dodecaedron to one of the bafes, is in power quintuple
to half the line which is between the planes.
6. And therefore the whole line which coupleth the
centers of the oppofite bafes is in power quintuple to
the whole line which is between the faid planes.
7. The line which fubtendeth the angle of the baſe of
the Dodecaedron, together with the fide of the baſe,
are in power quintuple to the line which is drawn from
the center of the circle which containeth the baſe, to the
circumference.
8. A fection of a ſphere containing three bafes of the
Dodecaedron, taketh a third part of the diameter of the
faid fphere,
9. The fide of the Dodecaedron and the line which
fubtendeth the angle of the Pentagon, are equal to the
right-line which coupleth the middle fections of the op-
pofite fides of the Dodecaedron.
THE
[ 337 ]
THE
THEOREMS
O F
ARCHIMEDES.
Concerning the Sphere and Cylinder,
inveſtigated by the Method of indivi-
fibles, and briefly demonftrated, by the
Reverend and Learned Dr. Is A A C
BARROW.
T
HE main Defign of Archimedes in his Treatife of
the Sphere and Cylinder, is to refolve theſe four
Problems.
1. To find the proportion of the fuperficies of a ſphere to
any determinate circle; or to find a circle equal to the fuper-
ficies of a given sphere.
2. To find the proportion of the fuperficies of any fegment
of a sphere to any determined circle; or to find a circle equal
to the fuperficies of any affigned fegment.
3. To find the proportion of the ſphere it ſelf (or of its
folid content) to any determinate Cone or Cylinder ; or to
find a Cone or Cylinder equal to a given Sphere.
४
4. Te
[ 338 ]
4. To find the proportion of a ſegment of a sphere to any
determinate Cone or Cylinder; or to find a Cone or Cylinder
equal to a given fegment.
Theſe four Problems Archimedes profecutes feparately,
and lays down Theorems immediately fubfervient to their
folution; but we reduce them to two: For fince an
Hemifphere is the fegment of a ſphere, and the method
of finding out its relations, in refpect to the fuperficies
and folid content, is comprehended in the general me-
thod of inveſtigating the proportion of the fegments:
And from the fuperficies and folid content of an Hemif-
phere already found, the double of them, (that is the
fuperficies and content of the whole fphere) is at the
fame time given. And indeed 'tis fuperfluous and fo-
reign from the Laws of good Method, to inveſtigate
their relations diſtinctly and ſeparately; ſo that if it were
not a crime, I might on this account blame even Ar-
chimedes himself.
The whole matter therefore is reduc'd to theſe two
Problems.
1. To find the proportion of the fuperficies of any fegment
of a sphere to a determinate circle; or to find a circle equal
to the fuperficies of a given fegment.
2. To find the proportion of the folidity of any fegment of
a ſphere to any determinate Cone or Cylinder; or to find a
Cone or Cylinder equal to an affign'd fegment of a sphere.
I shall refolve thefe Problems by another much easier
and ſhorter method: In which the order being inverted,
first, I ſhall ſeek the folidity of a fegment, and from
thence deduce its fuperficies; a thing which is in my
judgment well worth obferving, and perform'd (as I
know of) by none.
First therefore, for finding the folidity of a ſegment,
I ſhall lay down two, commonly known and receiv'd,
Suppofitions, viz.
1. That a ſeries of magnitudes proceeding in Arithmetical
Progreffion from nothing (inclufive) or whofe common diffe-
rence is equal to the leaft magnitude, is fubduple of as many
quantities equal to the greatest: (i. e. fubduple of the pro-
duct of the greatest term and number of terms :) So that
if the fum of the terms be called x, the greatest term g,
and the number of terms n, then will
?
ng
,
or 2 % ■ #g.
The
[ 339 ]
The truth of this Propofition will eafily appear by ex-
preffing the ſeries twice, and inverting the order
O, as 2a, 3a, 4a.
4a, 3a, 2a, a, o.
For fo the difference always being equal to the leaft
quantity, 'twill be evident that each two correſpondent
terms taken together are equal to the greateſt term; and
alfo, that the feries taken twice is equal to the greateſt
term repeated as many times as there are terms, i. e.
the laſt term drawn into the number of térms.
We have in a triangle a very clear and eafy example
of this moſt uſeful Propofition, which is prov'd hence,
to be half a parallelogrom having the fame altitude, and
Standing on the fame bafe.
Suppofe the altitude AE (Plate IX. Fig. 9.) of the
triangle AEZ to be divided into parts indefinitely many
and fmall, AB, BC, CD, DE, and parallels BZ, cź,
DZ, EZ, drawn thro' the points of Diviſion; all theſe
proceed from nothing in an Arithmetical progreffion, and
confequently the fum of them all, (that is, the triangle
AEZ) is fubduple of the greateſt EZ drawn into the
altitude AE, by which the fum of the terms is exprefs'd,
that is fubduple of the Parallelogram EY, whoſe baſe is
EZ, and altitude AE,
But the illuftration of the Rule will conduce more to
our defign by inferring hence, That a circle is equal to
half of the radius drawn into the circumference, after this
manner. Conceive a circle to confift of as many con-
centric Peripheries as there are points or equal parts in-
definitely many and ſmall in the radius. Theſe Periphe-
ries, as well as their radii, proceed from the center
or nothing in an Arithmetical progeffion; and therefore
their fum, that is, the whole circle is equal to half the
greateft (or extreme circumference) drawn into the
number of terms, that is, the radius.
* z
After
[ 340 ]
After the fame manner we may ſuppoſe the ſector AEZ
(Fig. 10.) to conſiſt of as many concentric Arcs BZ, CZ,
DŽ, EZ, as there are points (or equal parts indefini-
tely fmall) in the radius AE, which Arcs, as their radii,
proceeding from a point or nothing in an Arithmetical
progreffion, the ſector alfo will be equal to half the ra-
dius drawn into the extreme Arc EZ. Which may be
made evident alfo after this manner: Let us fuppofe
the right line EY to be perpendicular to the radius
AE, draw the right-line AY, and from the points
B, C, D, of divifion in the radius, draw BY, CY,
DY, parallel to EY, and terminated at AY. Becauſe
EY: DY (:: rad. AE: rad. AD): Arc. EZ: Arc.
DZ, and EYEZ, then will DY Arc. DZ; and
in like manner will CY= CZ, and BY = ᏴᏃ .
Whence the triangle AEY Will be to the ſector
AEZ, that is,
EXEY (AEXEZ ) __fector AEZ· By
2
2
this means we collect that celebrated Theorem of Archi-
medes, that a circle is equal to a triangle whofe baſe is
equal to the radius, and altitude equal to the periphery
of the circle; and that without any infcription or
circumfcription of figures, by only fuppofing that the
Area or Superficies of the circle confifts of infinitely
many concentric Peripheries. Which method of Indivi-
fibles, (now first of all known to me) feems no leſs
evident (nay more evident) and perhaps leſs fallacious
than that wherein planes are fuppofed to confift of
rallel right-lines, and folids of parallel planes; as here-
after fhall be evident, when we fhall collect, by this me-
thod, the proportions of ſpheric and cylindric fuperficies
to one another, by knowing the folid content; and on
the other hand, the folid content, by knowing the fuper-
ficies, with admirable facility, and full fatisfaction in
thoſe things which are with difficulty obtained by pure
Geometry.
pa-
Let us fuppofe a series of quantities to proceed from o
(inclufive) in a duplicate Arithmetic proportion, that is,
0, 1, 4, 9, 16, &c. the ſquares of numbers in a fimple
Arithmetic progreſſion, 0, 1, 2, 3, 4, &c. And the
triple of this feries will always exceed the greatest term
multiplied by the number of terms; but the number of
terms increafing, the proportion continually approximates,
till at last it comes to an equality, when the number of
terms is increaſed in infinitum.
3 X
}
[ 34 ]
3×0+1=3. 3
2X I=2. 2
3x0-1-1-+4=15
15 5
11
3X4—12. 12 4
3×OTI+4+9=42. 42 7
11
4X936. 36
36 6
3x+1+419+16=90. 90 9
5×16—80. 80 8
3×0+1+4+9+16+25=165.165 11
6X25=150. 150 10
As for example, if the terms are two, the triple of the
terms will be to the greateſt term drawn into the number
of terms as 3 to 2; if there be three terms as 5 to 4;
if four, as 7 to 6; if five, as 9 to 8, and fo continually
So that the antecedents of thefe proportions, always mu-
tually exceed one another by the number 2 ; and fo every
antecedent its confequent by 1. Whence it is evident
that by how much the greater the number of terms is,
by fo much the more the proportion tends to equality.
So 100 to 99 is lefs diſtant from the proportion of equa
lity than 10 to 9. From hence, fuppofing the number
of terms infinite (or infinitely great,) the triple of quan-
tities proceeding thus in a duplicate proportion (or as the
fquares of the numbers, o, 1, 2, 3, 4, &c.) will be
equal to as many quantities equal to the greateft term.
The fame, as to the fubftance of it, is laid down by
Archimedes in his Book of Spirals, as the Foundation of
many Argumentations, in that, and other Books, and is
well demonftrated by our learned Country-man Dr. Wallis:
However, I thought fit to illuftrate the matter by this
method, as being not unworthy our Confideration, and
very perfpicuous and intelligible in this, that 'tis free
from Fractions: And by the way 'tis obferv'd, that from
hence we may eaſily find the proportion of a ſeries triple
to as many terms equal to the greateft, viz. as twice the
number of terms lefs one, to twice the number of terms
lefs two. So that if the number of the terms be 6, the
proportion of a ſeries triple to as many terms equal to
the greateſt will be as 11,
Y 3
it
[342]
It will be a very eaſy and apt Illuftration of this Rule,
if we infer hence, That a Cone is fubtriple of a Cylinder,
having an equal bafe and altitude. For let us fuppofe
the altitude AE (Fig. 11.) of the Cone ZY to be divided
into equal and indefinitely number of parts, by as many
parallel right-lines ZY, and the lines ZY will be as the
numbers 1, 2, 3, 4, &c. and the fquares or circles
conftituted upon the diameters ZY, as 1, 4, 9, 16, &c.
whence all thofe circles, or the whole Cone AZY
(made up of the fame) will be fubtriple of as many
circles equal to the greateſt, conftituted on the greateſt
diameter ZEY, that is, fubtriple of a cylinder whofe
bafe is AEY, and altitude AE,
There occur two other moſt apt examples of this
Rule, viz. by inferring, That the complement of a Sea
miparabola is fubtriple of a parallelogram having the
fame bafe and height; as alfo, That the Space comprehended
by the Spiral and Radius is fubtriple of the circle in which
the Spiral is generated: But of thefe in another place.
Wherefore to go on with what we began, theſe two
Rules being fuppofed; let us conceive ZAY (Fig. 12.)
to be a fegment of a ſphere, X its center, AT its
diameter, and ZAYT a great circle paffing thro' the
vertex, and the part AE of the Axe to be divided into
an indefinitely number of equal parts; and let us imagine
parallel-lines to be drawn thro' the points of divifion
generating circles in the ſphere, whofe Radii let be
ĚZ, CZ, DZ, and diameters ZY. I fuppofe the feg-
ment of a ſphere to confift of all theſe parallel circles,
whofe number is as great as that of the points, or equal
indefinitely ſmall parts in the Axe AE, according to the
known Method of Indivifibles,
But nor for brevity's fake, let the diameter AT be
called d, and the radius of the ſphere r, (if need be) and
the Axe AE, by which the number of terms is exprefs'd,
calln, and one of the equal parts a; which being fet-
tled, 'tis evident, (by the Elements) that BZ AB
xBTaxda =
a× d— a — ad —a², and in like manner CZ²
ACX CT=2aXd—z azad.
zad — 4 a
4 a², and by the
× DT — 3 ad — 9 a², and
fame reaſoning DZ³ — AD
EZ AEX ET4 ad 16 a, &c. that is, that
ᎬᏃ
[ 343 ]
the fquares of the radii of the circles ZY are to one ano-
ther as the rectangles, ad, zad, zad, 4ad, &c. (which
proceed in an Arithmetical Progreffion from o) lefs by the
fquare a², 4a², 9a², 16a², &c. which go on
as the
fquares of the number, 1, 2, 3, 4, &c. But by our
firſt Rule, all the Rectangles o, ad, zad, zad, 4ad, &c.
are equal to half as many terms equal to the greateſt AE
nd X n
X AT or nd, that is,
2
Moreover, by our fecond Rule, all the fquares o, a³,
4a², 9a², 16a², &c. taken together, are equal to a
third part of as many terms equal to the greateſt AE² ΟΙ
o
n2 X n
n², that is,
3
Wherefore all the fquares defcribed upon the radii
BZ, CZ, DZ, EZ, conjunctly, are equal to the diffe-
rence
nnn
nnd
(or the terms being reduc'd to the
3
3ndn-2 nnn
fame denomination,)
and their quadruple,
6
6 ndn—4 n³
are equal to
or
whence a
6
3
that is, all the ſquares defcribed upon the diameters ZY,
12 ndn 8 n³
fegment of a ſphere is equal to a Cylinder, the diameter
of whoſe baſe is the fide of a ſquare equal to 6 nd—4n²
and altitude is n; or to a Cone having the fame baſe,
but the altitude n, or which is all one, having a bafe
6 nd-4 n²
whofe radius is v
or √√¾½ nd— n², and al-
titude nas before. Which Cone we may change into a
Cone upon the fame baſe ZY with the fegment ZAY, by
faying, as ZE² (i. e. dn -n²) to 2nd-n² or (both
terms being divided by n) as d-n to d-n, fo is recipro-
callyn to the altitude of the Cone fought: Or in the
figure by making, as TE to TE + XA, fo is EA to ES.
For ES will be the altitude of the Cone ZSY equal to the
Segment of the Sphere ZAY. Which is that noted Theorem
of Archimedes, demonftrated by him with fo much labour
and prolixity.
Y 4
Hence,
[344]
Hence, if the given fegment be a Hemisphere, and fo
nd or r, then dor 2 r will be the altitude of a
Cone, which having a bafe equal to the bafe of the He-
misphere (or to the greatest circle in the ſphere) will be
equal to the Hemisphere. And a Cone whoſe baſe is double
of the greateſt circle, and the altitude 2 r, or the Cylinder
whoſe baſes is of the greateſt circle, and altitude 2 r
will be equal to the whole Sphere. Whence the whole
Sphere is of a Cylinder, the diameter of whofe bafe is
2 r, and the altitude alfo 2 r. And this is the chief
Theorem of Archimedes, viz. That a sphere is fubfefquialter
or of that Cylinder, whofe Altitude and Diameter of the
baſe is equal to the Diameter of the Sphere.
2
3
Furthermore, not to pass over any thing in our Au-
thor which feems to be to our purpoſe:
If to the fum first found, reprefenting a fegment,
6 ndn 4 nnn,
viz.-
3
2
2dn- 6dn²+4x³
we add
3
4dn—4 nå.
3
d-zn
X ≈4 ZE²× XE) repreſenting the Cone ZXY,
2
I
44
the aggregate dd n will repreſent the Sector of the
Sphere ZXYA, which for that reaſon will be equal to a
Cylinder, the diameter of whoſe baſe is dn, and the
altituded, or to a Cone, the diameter of whoſe baſe is
✔ dn, and the altitude zd, or alfo to a Cone, the Ra-
dius of whoſe baſe is ↓↓ dn, and the altitude ½ d — r (it
being reciprocally as 4 dn: dn: : 2d: d,) that is, to a
Cone, the Radius of whoſe baſe is the Line AZ, drawn
from the vertex to the circumference of the baſe of the
fegment, (for AZ TAX AEdn,) and the altitude r.
And this is the next famous Theorem of Archimedes, con-
cerning the folidity of the ſector of the Sphere, viz, That
the ſector of a ſphere is equal to a Cone, whoſe baſe is a
circle defcribed by a Radius equal to a line drawn from
the vertex to the circumference of the base of the Segment,
and whofe altitude is equal to the Radius of the Sphere.
}
And thus I think I have compleated that which be.
longs to the folidity of a ſphere, and its parts, with ſuf-
ficient brevity and perfpicuity. From hence we fhall
deduce the Refolution of the other Problem, which I
propoſed concerning the furface of the fegment of a
fphere; and then of the whole fphere. To obtain this,
as we fuppofed before, a Circle to conſiſt of concentric
Peripheries,
!
[ 345 ]
Peripheries, and the Sector of a Circle of concentric Arcs,
(in the number of which, the greateſt, and the leaſt,
or a point is reckon'd: So now we fuppofe fpheres to
confift of concentric ſpherical fuperficies, and the Sector
of Spheres of like concentric fuperficies; as for example,
the fector of the ſphere ZAE (Fig. 13.) of the fuperficies
BZ, CZ, DZ, EZ, &c.) which fuppofition indeed
ſeems ſo eaſy and natural, that in my judgment 'tis
fufficient only to propofe it; neither is a further expli-
cation wanting to gain an affent to it.
2. We fuppofe thefe fpherical fuperficies to be in a
duplicate Ratio of the Radius of the ſpheres: This is the
common affection of all like fuperficies, and it ſeems to
agree very well with the fuperficies of fpheres, becaufe
they appear to be moſt uniform and fimilar. But this
Suppofition might eafily be evinc'd and eſtabliſh'd by the
fame fort of arguing, as fpheres are proved to be in tri-
plicate proportion to their Diameters or Radii; or might
have been join'd as a Corollary to Prop. 17.and 18.Elem. 12.
where the fuperficies of like Polygons are fuppos'd to be
infcribed in ſpheres, having as well the fuperficies in a du-
plicate, as the folidity in a triplicate Ratio of the Diam-
eters of the Spheres. Thefe things being premis'd let us
fuppofe AE a Radius, or the fide of the Sector of a Sphere
EAZ, to be divided into equal and indefinitely many
ſmall parts, and the ſector AEZ to confiſt of theſe ſpheri
cal fuperficies BZ, CZ, DZ, EZ, it will be evident that
all thofe fuperficies in the Progreffion are as the fquares
of the Radii, that is, as AB, AC², AD², AE¹, &c.
or as the ſquares of the numbers 1, 2, 3, 4, &c. whence
by our fecond Rule, the fum of all thefe fuperficies, that
is, the fector AEZ, will be of as many fuperficies equal
to the greateſt EZ, that is, of the greateſt EZ, drawn
into the number of terms. Whence a fector is equal
to a Cylinder, whoſe baſe is of the greateſt or extreme
fuperficies of the fector, and whofe altitude is r: Or to
a Cone whofe bafe is equal to the fuperficies of the fec-
tor, and its altitude r, which is the laſt of Lib. 1. but
we just now prov'd that a fector is equal to a Cone
whoſe altitude is r, and baſe a circle, defcrib'd by the
Radius YE, drawn from the vertex of the ſegment EYZ
to the circumference of the baſe. Wherefore a Cone,
whoſe altitude is r, and baſe equal to the fuperficies of
the fector, is equal to a Cone of the fame altitude, whoſe
bafe is a circle defcrib'd by the Radius YE.
And
1
[346]
And fo the fuperficies of the ſector EYZ is equal to a
circle defcrib'd by the Radius YE. Which certainly is
the principal Theorem of all thoſe that occur in the
Books of Archimedes, nor is there found a more excel-
lent one in all Geometry; viz. That the fuperficies of any
Segment of a Sphere is equal to a circle whofe Radius is a
right-line drawn from the vertex of the fegment to the cir-
cumference of the bafes: And hence, that the fuperficies of
an Hemisphere is double to the baſe, or equal to two great
cirles of the Sphere.
For in this Cafe YE AZ AY— 2 AE2, and
— --
confequently a circle deſcribed by the Radius YE (Fig. 14.)
is equal to two circles defcrib'd by the Radius AE.
Whence alfo, the fuperficies of the whole fphere is quadruple,
a circle having the fame Radius with the sphere, that is,
quadruple the greatest circle in the fphere; and equal to a
circle whofe Radius is the diameter of the sphere. From
hence it follows, that the fuperficies of a Sphere is equal to
the fuperficies of a Cylinder of the fame beight and breadth;
for the fuperficies of that Cylinder is quadruple to the
bafe as we ſhall fhew hereafter. And theſe are the
moſt noted Theorems of Archimedes. Nay, from hence
all thoſe things follow, which he has written concern-
ing the fuperficies of fpheres, and their fegments. So
that from thefe few and eafy Suppofitions, I have de-
monftrated whatever feems to be of any Note in the
Books of the Sphere and Cylinder.
I will only add, that after by the method of Archime-
des, (for I think ſcarce any other can be invented, befides
ours, for finding the folidity) the fuperficies of fegments
are found equal to the circle defcribed by the Radii
YE; hence it will plainly follow, that the fuperficies of
Spheres, and thence of like ſectors are in a duplicate ratio
of the Radii of the ſpheres; and confequently from the
fuperficies thus found, the contents of fegments, and of
whole fpheres may be mutually deduced, and that very
clearly and expeditiouſly, after this manner. Becaufe in
the ſector EAZ (Fig. 13.) the fuperficies BZ, CZ, DZ,
EZ, proceed as the fquares defcrib'd upon AB, AC,
AD, AE, that is, as 1, 4, 9, 16, &c. the whole ſector
will be equel to of as many fuperficies equal to the
greateſt EZ, or EZ × r, that is, to a Cylinder whofe
baſe is EZ, and altitude r, or to a Cone whoſe baſe
is EZ, and altitude r. But EZ is fuppofed equal to a
sircle whofe Radius is YE, wherefore the fector EAZ
is
[ 347 ]
is equal to a Cone whoſe baſe is a circle defcribed by
the Radius YZ and altitude r: Which is Archimedes's
univerfal Theorem for the contents of Sectors. Whence
3
if from this the Cone ZAE ftanding on the baſe of the
fegment EYZ, and having the vertex at the center of the
ſphere A, be fubducted, you'll have that fegment EYZ.
But when the ſector EYZ is a Hemiſphere, there will be
no fuch Cone to be fubducted; and for that reaſon a
Cylinder whoſe baſe is EZ, and altitude r, or the
Cone whoſe baſe is 2 EZ, and altitude likewiſe r, will
be equal to the whole fphere. But the Superficies of
the Hemiſphere EZ, is proved to be equal to two of the
greateſt circles in the ſphere, whence the whole ſphere
is given. This is Archimedes's firſt and principal Theorem,
for the content of a ſphere; whence 'tis eafily deduced,
that a ſphere is of a circumfcrib'd Cylinder, that is,
of a Cylinder whofe altitude and diameter of its bafe is
equal to the diameter of the ſphere.
2
The Doctrine of our author [Archimedes] feems to
makes againſt, and fubvert the new and celebrated Me-
thod of indivfibles, and is prefs'd to that end by Tacquet;
for inftance, (Prop. 2. lib. 2. Cylindr.) For the uſual pro-
cefs of that method feems to exhibit the dimenfions of
the fuperficies of a Cone, (as alfo of a fphere, and of
other Curves) different enough from what our Author
and others have demonftrated: As for example, let us
fuppofe ABCD (Fig. 15.) afright Cone, whofe Axe is
AX, bafe BCD, and plane bxd drawn, at pleaſure,
parallel to the baſe BCD. And fince, as Diam. BD :
Periph. BCD:: Diam. bd.: Periph. bxd, and fo every
where it will be (according to the Method of Indivi-
fibles, and by 12. 5.) as Diam. BD, to Periph. BCD,
fo is the triangle ABD, confifting of thofe parallel Di-
ameters, to the Conic Superficies ABCD, confifting of
thofe Peripheries, i, e. Diam. BD: Periph. BCD: :
AX X BD AX x Periph. BCD
2
AXX Periph. BCD
N.
Whence
will be equal to the fuperficies of
the Cone; which is falfe and contrary to what was de-
monſtrated juſt now. For we demonftrated that the
AB × Periph. BCD
fuperficies of the Cone was
I
IN
In
[ 348 ]
In anſwering this Objection, we fay, that the Method
of Indivifibles, in the fpeculation of Perimeters, and of
Curve Surfaces, proceeds otherwife than in the fpecula-
tion of plane Surfaces and folid Contents. It does in-
deed fuppofe that the Area of plane Figures confifts as
it were of parallel right-lines, and the contents of fo-
lids of parallel Planes, and that their number may be
exprefs'd by the altitude of the Figures: But it by no
means fuppofes, that the Perimeters of plane figures con-
fift of points, or the fuperficies of folids of lines, the
number of which may be exprefs'd by the altitude of
the figure. As for example, altho' the triangle ABD
(in the laſt figure) confifts of lines parallel to BD,
the number of which is expreffed by the number of points
in the perpendicular AX, that is, by the length of
the perpendicular: Yet it would be abfurd to fuppofe
that the line AB confifts of points, whofe number may
be exprefs'd by the number of points in a leſs line AX,
For altho' the right-lines b d drawn thro' each infi-
nitely fmall part of AX, divide AB into as many infi-
nitely ſmall parts, yet thoſe parts are not of the fame
Denomination or Quality with the parts of AX, but
fomewhat greater than them; fo that if the parts of
AX be look'd upon as points, the parts of AB are not
to be called points, but greater than points; and on
the contrary, if the parts of AB be called points,
the parts of AX are to be look'd upon as less than
points, if it be lawful to ſpeak fo. For the points
which are treated of in the Method of Indivifibles are
not abfolutely points, but indefinitely fmall parts,
which ufurp the name of points, becauſe of their
affinity to them. Since therefore points don't admit
of greater and lefs, the name of points is not at the
fame time to be attributed to the parts of different mag-
nitudes; confequently, tho' the number of the greater
parts of AB may be exprefs'd by the number of the
Teffer parts of AX, yet the number of points în AB
can no ways be expreffed by the number of points in
AX, (that is, by the number of parts in AX, equal to
the number of parts in AB, which are called points:)
The line AB has as many points as there are in it ſelf
alone, or another line equal to it felf, nor can it be
determined by any other meafure. After the fame
manner, this method don't fuppofe the conic Surface
ABCD to confift of as many parallel circumferences
perpetually
[ 349 ]
perpetually increafing from the vertex A, or decreaſing
from the baſe BD, as there are points in the Axe AX,
but rather of as many thus increafing or decreafing as
there are points in the fide AB. For in the Revolution
of the line AB about the Axis AX, (whereby the ſuper-
ficies of the Cone is generated) every point in the line
AB produces a circumference, and confequently more
circumferences are produced than the points contained
in the Axis AX. Therefore if you would extend the
Method of Indivifibles to the fuperficies of folids, and
ſuppoſe thoſe fuperficies to confift of parallel-lines, you
ought not to compute this by the parallel Areas confti-
tuting the folids, that is, not to number thoſe Areas by
the altitude of the folid, but by other lines agreeable to
the condition of each figure. Which lines, in figures
that are not irregular, may eaſily be determined: For
inftance, in the equilateral Pyramid ABCD, (Fig. 16.)
whoſe Axe is AX, fuppofing that the lateral furface of
the Pyramid confifts of Perimeters of triangles, parallel
to the baſe BCD, theſe can neither be computed by
the altitude AX, nor by the fide AB, (for by the former,
the thing required would be wanting of the true Di-
menfion, and by the latter 'twould exceed it) but by
the line AE drawn from the vertex A pependicular to
the fide BC of the baſe: The reaſon of which is,
that every plane fide of a Pyramid, as ABC, confifts
of parallel right-lines computed by the altitude AE.
After the fame manner, fuppofing that the fuperficies of
the Hemiſphere BAD, (Fig. 17.) confifts of Peripheries
of circles parallel to the bafe BCD, the number of them
is not to be computed by the Axis AX, but by the
Quadrantal Arc AB, becauſe that every point of the Arc
AD in revolving produces a circumference; and fo any
fuperficies, whether plane or curv'd, which is conceived
to confift of equidiftant right or curv'd lines, is to be
computed by a line cutting thofe equidiftant lines per-
pendicularly. For fince thofe equidiftant lines, in this
Method of Indivifibles, are not confidered abfolutely as
lines having an infinitely fmall breadth, which is the
fame with the breadth or thickneſs of the point de-
fcribing thoſe equidiftant lines in their Circumvolution,
and fince the fame equidiftant lines divide the line
cutting them perpendicularly into parts meaſuring its
breadth, thofe parts are to be looked upon as fuch
fort of points, and confequently the number of equi-
diſtant
[350]
diftant lines, or the fum of thofe breadths is to be
computed by the number of points in the line cut-
ting them perpendicularly, that is, by the length of
that line, and not by a line of any other length, for
that will confift of more or less points.
Hence therefore in the fpeculation of the fuperfi
cies of folids, the Method of Indivifibles is not unufeful,
but rather very commodious, provided it be rightly
underſtood, and applied according to the Rule pre-
fcribed. For by the help of it even thefe fuperficies
may be found, if we have fome convenient Data pre-
fuppofed, on which the reafoning may be founded:
For inftance, we might, by the help of it, inveſtigate
the fuperficies of a Cone, by reafoning after this
manner.
If the fuperficies of the Cone ABC (Fig. 15.)
be divided into innumerable Peripheries of circles
bxd parallel to the bafe BCD, the breadth of
thoſe Peripheries taken together, make up the fide
AB cutting them perpendicularly, and confequently
there will be as many Peripheries as there are points in
the line AB, that is, their number may be exprefs'd
by the number of points in AB, or by its length.
Wherefore if you draw perpendiculars equal to the
Peripheries to every point of AB, a fuperficies will be
made out of thofe perpendiculars equal to the fuperficies
of the Cone; but that fuperficies will be a triangle
whoſe height is AB, and bafe equal to the greateſt
Periphery BDC, and fo the fuperficies of the Cone
will be AB x Periph. BDC, which conclufion
agrees with the things laid down and demonſtrated
by Archimedes.
After the fame manner, if you take any right-line
ab (Fig. 18.) equal to the quadrantal Arc AB of the
Hemiſphere (Fig. 17.) and to each of its points m let
the perpendiculars m n be erected equal to the Radii
MN of the parallel circles MOM paffing through the
correfponding points M of that quadrantal Arc, the
greateſt of which bx let be equal to the Radius BX
of the bafe of the Hemifphere: The figure a bx will
contain the Radii of all the circles of whofe Periphe-
ries the fuperficies of the Hemiſphere confifts. And
if the perpendicular mo, b d be erected equal to the
Peripheries MO, BD, there will be made the figure
a b d
[351]
A
bd equal to the fuperficies of the Hemiſphere. The
dimenfion of which figure if you can by any means
find (as in this cafe you are to find the Area of the
figure abx) thence you will eafily deduce the content
of the fegment of the fphere, agreeing to what you
would gather by any other genuine method of rea-
foning. Which Obfervation, I think, will not be
unufeful in Geometry.
AN
A N
APPENDIX
CONTAINING
The NATURE, CONSTRUCTION,
and APPLICATION
O F
LOGARITHMS.
L
Ogarithms are a ſet of artificial numbers fo propor-
tioned among themſelves, and adapted to the natural
numbers, fo as to perform the fame things by addition
and ſubtraction, as thefe do by multiplication and divifion.
From this definition it follows, 1. That in any ſyſtem or
table of Logarithms whatever, the Logarithm of unity or 1,
will be nothing for as one neither increaſes nor diminiſhes
the number multiplied by it, fo neither will its Logarithm
either increaſe or diminiſh the Logarithm to which it is ad-
ded; and confequently the Logarithm of 1 muft be nothing.
2. For a like reafon, the Logarithm of a proper fraction
will always be negative; for fuch a fraction always dimi-
niſhes the number multiplied by it, and therefore its number
will always diminiſh the Logarithm to which it is added.
I
3. This property of Logarithms, affords us no ſmall
compendium in multiplication; for whenever one number
is to be multiplied by another, it is but taking out their
Logarithms, and adding them together, and their fum will
be a third Logarithm, whofe natural number being taken
out of the tables will be the product required.
4. The fubtraction of Logarithms anſwers to the divifion
of natural numbers to which they belong; that is, when-
ever one number is to be divided by another, it is but
fubtracting the logarithm of the divifor from the Logarithm
of the dividend, and the remainder will be the Logarithm
Z
of
354
An APPENDIX.
of the quotient; and bccaufe every fraction is nothing elfe
but the quotient of the numerator divided by the deno-
minator, its Logarithm will be found by fubtracting the
Logarithm of the denominator from the Logarithm of the
numerator. To demonftrate this let the number x be di-
vided by the number y; and let the quotient be the
number z; and let the Logarithms of the numbers x, y, z,
b, c, respectively; I fay then that a→→→ bc; for
fince by the ſuppoſition x, we fhall have x-
be
а,
X
abc by the definition; whence a - b C.
yx, and
5. As every fourth proportional is found by multiplying
the fecond and third numbers together, and dividing the
product by the firft, fo the Logarithm of every fuch fourth
proportional will be found by adding the Logarithms of the
fecond and third numbers together, and fubtracting from the
fum the Logarithm of the firft This renders all the ope-
rations by the rule of proportion very compendious and
eafy; this compendium is chiefly uſeful in Trigonometry,
both plain and fpherical, which it renders very eafy to be
performed.
6. If x be any number, whofe Logarithm is a, then the
Logarithm a² will be 2a, that of x³, 3a, &c. that of
a,
I
that of
x²,
2
>
Σ
X
2a, &c. And univerfally, the Loga-
rithm of am will be a Xm, and that, whether the index m
be intregal or fractional, affirmative or negative on the
other hand if be the Logarithm of any power of x, as of
9
9
x", then will be the Logarithm of x. The reafon of all
m
ہو
this is plain; for as is the product of x multiplied into
itfelf, fo its Logarithm will be the Logarithm of x added
to itself or doubled, that is za; and ſo of the higher powers.
I
Again, as is the quotient of unity divided by x, its Loga-
rithm will be found by fubtracting a from o, the Logarithm of
, which gives-a; and fo of the other powers. Laftly,
as√x, when multiplied into itſelf produces x, fo its Loga-
rithm when added to itſelf ought to make a; therefore the
Logarithm of √x will be a, and fo of all the other
fictional powers.
7. If any fet of numbers 1, x, xx, x³, &c. be in conti-
nual geometrical progreffion, their Logarithms, o, a, b, c,
&C,
An APPENDIX.
355
ذ
&c. will be in arithmetica! progreffion: For fince by the
fuppofition is to x, as x is to xx, as xx is to x³; that is,
x3
we ſhall have a-o- =b
X
xx
fince
I
x
3
DTX
a
b by the fourth confectory; therefore o, a, b, c, are in
arithmetical progreffion.
In the progreffional feries of geometrical proportionals, if
between the terms 1 and x be inferted a mean proportional,
which is x, its index or Logarithm will be, becauſe
its diftance from unity will be but one half of the diſtance
of from unity, and confequently the root of x will be
expreffed by . In like manner, if between x and x² we
infert a mean proportional, its index will be 1 and or 2,
becauſe its diſtance from unity is once and half the diſtance
of x from the fame place of unity.
3
2
2
3
Again, if between 1 and be inferted two mean pro-
portionals; the firſt of theſe will be the cube root of x or
√³x, and its index will be, becauſe its diſtance from the
units place is but one third of the diſtance of x from the fame
place of unity; and confequently √3x is expreffed by x³;
whence it follows, that the index, or Logorithm of unity
is nothing, as we have before obſerved; becauſe unity
cannot be removed at any diſtance from itſelf.
And the fame ſeries of geometrical proportionals may be
continued on the contrary fide of unity, or towards the
left-hand, and which will therefore decreaſe in the fame
proportion as the terms placed on the right-hand increaſes.
For the terms
20
I
I
x
I
x3
Ι
I
x
- 1; x, x², x³,
x4, x³, &c. are in the fame geometrical progreffion con-
tinued, and have all the fame common ratio; therefore
becauſe the diſtance of x from unity, towards the right-hand
I
is pofitive, or + 1, the diſtance of from unity towards
x
the left-hand, which is equal to the former, is negative, or
I, confequently the index of
I
X
may be written x
Alſo, becauſe x2 is on the right fide of unity, and its
index+2, the index of
I
+2
و
which is as far diſtant from
unity on the contrary fide, or towards the left-hand, will
I
bez, and confequently
is the fame with x
In
x
Z 2
like
356
An APPENDIX.
like manner x
I
3
I
4
is the fame with
and x
the fame
with ; whence it follows that thefe negative indices
34
ſhew that the terms to which they belong, are as far diftant
from unity towards the left-hand, as the terms whofe indices
are the fame and pofitive, are removed from unity on the
contrary fide, or towards the right-hand.
4
Again, if between 1 and x, in the feries increaſing from
unity there be inferted a mean proportional as √x, and
between this mean proportional, and unity be inferted
another mean proportional, as x, and between this
laft mean proportional and unity be inferted another mean
proportional asx, &c. thofe refpective indices will be
1, 1, 1, &c. likewife if between unity and the next term
I
decreafing be inferted a mean proportional, as,
I
X
8
Ι
√x
the
correfpondent index will be ; and if between this mean
proportional and unity be inferted another mean proporti-
I
onal as 4
་
Jo
the correfpondent index will be
1, &c. And
4
as the fame may be done between any other two terms of the
progreffion, it follows, that between any two terms may be
inferted an infinite number of mean proportionals, whoſe
reſpective indices will become the Logarithms of the re-
ſpective terms to which they belong: And theſe mean
proportionals the greater they are in number, between the
two terms of any ratio, the nearer they approach to a ratio
of equality, with their correfpondent indices or exponents.
Hence we ſee the reaſon of the ancient method of making
Logarithms, and which was practiced by their firſt inventors,
which was, to extract the fquare root, out of the fquare root,
&c. of any number, in order to find a ſeries of continual mean
geometrical proportionals; till the number of cyphers contain-
ed bewteen unity, and the firſt ſignificant figure of the root,
was equal to the number of places the intended Logarithm.
fhould confift of; and at the fame time to find out a like num-
ber of correſpondent arithmetical means, which would be the
Logarithms of the geometrical means refpectively.
But how tedious and opperofe this method is, may be
eafily imagined, by any one who has been at the pains to
extract the root of a large number; and may be gathered
from this, that to find a Logarithm to feven places only,
requires
An APPENDIX.
357
requires at leaſt 27 extractions of root out of root, the ſquare
conſiſting of 16 figures at leaft; and if in any one of the ope-
rations an error happens, the whole work muſt be repeated.
From what has been faid we may obferve that the
Logarithm of any number, is the Logarithm of the ratio
of unity, to the number itſelf; or it is the diſtance between
unity and the fame number, in the geometrical ſcale of
proportionals, and is meaſured by the number of proporti
onals contained between them: Logarithms therefore ex-
pound the place or order that every number obtains from the
units place in the continued feries or fcale of proportionals,
infinite in number.
Thus, if between unity and the number 10, there be
fuppofed 10,000,000, &c. mean proportionals, that is, if
the number to be placed in the 10,000,000th the place from
unity, between 1 and 2 there will be found 3010300 of
fuch proportionals; that is, the number 2 will be placed
in the 3010300th place from unity; between 1
and 3,
there will be found 4771213 of the fame proportionals, or
the number 3 will ftand in the 4771213th place, in the
infinite fcale of proportionals, which numbers 10000000,
3010300, 4771213, are therefore the Logarithms of 10, 2,
and 3; or, more properly, the Logarithms of the ratio's
of thoſe numbers one to the other.
Again, if between unity and the number to there be
fuppofed an infinite fcale of mean proportionals, whoſe
number is 23025851, &c. that is, if the number 10 be
placed in the 23025851ft place from unity, in the infinite
fcale of proportionals between 1 and 2 there will be found
to be 6931471, &. fuch proportionals; and between ▾ and
three 10986122, &c. of the fame proportionals; ſo that
if the firſt term of the feries be called x, the ſecond term
will be x2, the third x3, &c. and if the number 10 be
fuppoſed to be the 10000000th term in the feries, the number
2 will be the 3010300th term, and the number 3 will be
the 4771213th term in the fame feries; and confequently
x 1000000 10,000
=2, and x47 7 1 2 3
=3, &c.
and again, if the number 10 be fuppofed the 23025851
term, in the ſeries, the number 2 will be the 6931471 ft
term, and the number 3, the 10986122nd term in the fame
feries; and confequently, in this cafe, x2302585110,
XT93 147 1 2, and 8 6 1 2 2
= 3, &c.
co
C
And as the infinite number of mean proportionals between
any two numbers may be affumed at pleaſure, hence Loga-
rithms may be of as many different forms (that is, there
may be as many different fcales of Logarithms) as there
23
can
358
An APPENDIX.
can be affumed different fcales of mean proportionals be
tween any two numbers, Every number therefore is fome
certain power of that number which is placed next to unity;
in the infinite fcale of proportionals between unity and the
given number, and the index of that power is the Logarithm
of the number.
Logarithms therefore may be of as many forts as you
can affume different indices of the power of that number
whofe Logarithm is required,
And becauſe the ratio of unity to any number is meaſured
by the number of ratiuncula contained in that ratio; we
may therefore value ratio's by the number of ratiuncula
contained in each ratio, and may confider them as Quantates
fui generis, beginning from the ratio of 1 to 10, being
affirmative when the root is increaſing, but negative when
decreasing.
So that ratio's are to each other as the number of like
and equal ratiuncula contained between the two terms of
the ratio; wherefore, if the ratio of 1 to 10 contain
10000000, &c. equal ratiuncula, that of its duplicate, or
of 10 to 100, will contain 20000000, &c. that of its
triplicate, or of 100 to 1000, 30000000, &c. of the fame
fimilar and equal ratiuncula; fo that the Logarithms of
numbers, or the values of ratio's, are in an arithmetical
progreffion; and hence arifeth the common definition of Lo-
garithms, viz. that Logarithms are a ferics of numbers in
an arithmetical progreffion fitted or affigned to a rank of
numbers in a geometrical progreffion.
If the diſtance between unity and any number be ſuppoſed
to confiſt always of the fame number of mean proportionals
as 10000000, &c. or, which is the fame thing, if we make
the ratio of unity to any number, to confiſt always of the
fame infinite number of ratiuncula, the magnitudes of each
in this cafe, will be, as their number in the former; fo that
if between unity and any number propofed, there be taken
an infinity of mean proportionals, the infinitely fmall aug-
ment or decrement of the firft of thoſe means from unity,
will be a ratiuncula; and becauſe this is always proportional
to the momentum or fluxion of the ratio of unity to the
faid nummber; and fince in theſe continual proportionals all
the ratiuncula are equal, their fum, or the whole ratio, will
be as the faid momentum or fluxion is directly; wherefore if
the root of the infinite powers be extracted out of any num-
ber, the difference between that root and unity, will be as
the Logarithm of that number; fo that the infinite root of
any number being found, the Logarithm of that number
is
An APPENDIX.
359
is cafily computed. How the infinite root of any number
may be found, muft be next confidered.
Let p+px be a given quantity whofe n root is to be ex-
tracted, where n ftands for any power or number taken
at pleaſure.
-n
12
・n
Then will p+ px = px 1 + x.
12
Aſſume 1+x=1+ Ax+Ba² + Cx³ -|- Dx² -|- Ex³, &c.
Then "x1+x
Ex+ x, &c.
And x1+x
πx1+x
I
·n- -I.
NI
NI
X
— Ax+2Bxx+3Cx²x+4 Dx ³ x+5
(Ex¹, &c.
=A1-2 Bx + 3 Cx² + 4 Dx³ +5
(Ex4, &c.
A+ 2 Bx + 3 Cx² + 4 Dx³ + 5
And-
n
1 + x
1 + Ax+ Bx² + Сx³ + Dx² +
Ex¹, &c.
Alfon A, and nA
2 BA.
And лA — A: 2 B.
And n-n Ax + n B x ² + n Cx ³ -|- n Dx+, &c. —A+ 2 Bx
+ 3 Cx² + 4 Dx² + 5 Ex¹, &c.
Alſo n B = 3 C+ 2 B.
And ʼn B
n
2 B
3 C.
And n
IXA2 B.
And n -
2 X B = 3 C.
72
n
And
མ
× A= B.
And
2
2
Alfo nC 4 D+ 3 C,
==
And C3 C4D,
n
And n
22 2
And
4
3x C—4 D, .
=
XC = D.
न
3
X B — C.
Alfo D5 E+ 4 D.
12
And nD
5 E.
4X D = 5 E.
4 D
And n-
ท
And
4x D = E.
5
ท
Wherefore A—n, В — ~ X
C=nx
X
2
2
n
2.
n
n
2
n
2
,
D=nX
X
X
and En x
n
3
2
X
2
3
4
22
2
X
X
3
4
n
n- 3 2 - 4
5
&'c.
ZA
And
360
An APPENDIX
n
And, 1+x=1+ nx + n x −
#22
n
3
1 n
X
2 n
3
X
3
2
4
n
And p+px=" + n p" x + nx
p+px=p”
n
カー
​71-1
I 11
17 2
n
p x³, &c.
x+, &c.
2
x
N.
2
n
p" x² + " x
2 3
n
n
Put p" A, np" B, nx
n
2
3
p² x3 = C.
-n
2
Then p+ px =p" + n Ax+
cx+
5
4
12
px² = C, nx
=C,
n
n
Bx+
2
X
2
3
Ex, &c. Which is the celebrated binomial
feries invented by the illuftrious Sir Ifaac Newton, for raifing
a given quantity to any given power, or extracting any
root out of the fame quantity; "becaufen ftands for any
number whether it be a whole number or a fraction.
nis finite, 1+x
1
I
I
I - N
1-311+212
n
+
-nx+
*2
x² +
6n3
6n3
6 n + 11 n²
2424
When
3
2 nn
x² +
*+, &c. but when it is infinite
I
I
I
I
n
=i+
12
2n
372
I
x+-----x³, EP°c,
4n 5n
nn being infinitely infinite, and confequently whatever is
1
divided by it vanifhes; whence it follows, that multiplied
into x
1/2 x 2 + 1/1/ 3
x
I
3
I
x4, &c. is the augment of the
firft of the mean proportionals between unity and 1+x, and
is therefore the Logarithm of the ratio of 1 to 1+x.
But becauſe the infinite root of 1
I
1.
•n
I z, or 1 * is
*—*—*—*— —*—*—*, &c. there-
I
3
37
I
I
1
78
218
372
42
>
fore
An APPENDIX.
361
fore
I
2
3
× into x+x²+ ÷ ׳ + \ x4 -↓-x, &c. will be the
Logarithm of the ratio of 1 to 1-x, or the Logarithm of
a number lefs than unity; and whereas the infinite index n
may be taken at pleaſure, the feveral fcales of Logarithms
to fuch indices will be reciprocally as
indices, and if the index be taken
refpective Logarithms will be fimply,
2n
I
+-x x3 +
372
I
4n
1
n
> or as the ſeveral
10000000, &c. the
I
x4 --— x³, &c.
5n
the
Let a repreſent the leaft of any two given numbers, b
the greateft, the fum of the two numbers, and
difference, and let us ſuppoſe the ratio of a to b, to be di-
vided into that of a to z, and of z to b; that is into
the ratio of a to the arithmetical mean between the two
numbers, and the ratio of the faid arithmetical mean to
2
the greater number b; then becauſe
a
a
X =
b b
the
a
12
a
Logarithm of +Log.
b
= Log. of
> or
becauſe
b
b
1%
b
b
the Log. of
+Log.
Log. ; that
a
2x b
X
a
is, the fum of the Logarithms
Log. of the ratio of a to b;
theſe ratio's in the terms of
a : : 1 : 1 − x, whence x-
1
I
a
a
of theſe two ratio's will be the
and to find the value of each of
and I +xwe muſt ſay, as ½ z :
b - 1/2 % d
42242
Z
Ι
Z
and fubftituting -
≈
in the room of x, in each of the former feries, we fhall have
I
1
1
X
d
dr
d3
+
+
Z
22 2
323
22 1
dr
ds
+ + &c. for the Log.
of the ratio of a to z, and
24
ds
4x+
+
5x5
424
5x5
1
I
X
22
Z
א
d2
d3
+
222
323
&c. for the Log. of the ratio to b,; and
confequently the fum of thefe two feries
I
2d
2d3
X
+
22
Z
+
323
ds
d7
+
&c.
285
d 13
&c. or
X 2 X
525
22
+ +
2 323 525 727 '929
will be the Logarithm of the ratio of a to b, whoſe
difference is d, and fum x; whence to find the Logarithm
of any prime number we have this general rule,
To
362
An APPENDIX.
ઃઃ
CC
"To the given number add 1 for a denominator or
divifor, and fubtract 1 from the fame number for a
"numerator or dividend, then of the vulgar fraction thence
refulting, compofe all the odd powers thereof, theſe will
"form a ſeries of numbers which being divided by their
respective exponents, viz. 1, 3, 5, 7, 9, &c. will produce
a feries of quotes, whofe fum will be the natural Loga-
"rithm of the number propofed, and being doubled, will
give the Logarithm of the fame number, according to
Neper's form.
CC
>>
This rule is not only very eaſy to be retained in memory,
but very proper for the practice of making Logarithms, which
it performs with the greatest expedition, and is fo very plain
that it may be taught to perfons of a mean capacity.
For example, Let it be required to find the Logarithm of
2, the firft prime number.
Becauſe a 1, and b2, therefore d — b
and ab
I
Hence +
II
I
Į
3
3
+ 81
=3; wherefore
I
X
1
I
27
રર
a = 1,
d
d2
=
and
9
I
I
I
I
+
X
+
X
&c.
S
243
7
2187'
I
+ +
1215 15309
2
will be the natural Loga-
2
2
rithm of 2, and confequently + + +
2
3 81 1215 15309
&c. will be the Logarithm of the fame number in Neper's
form; wherefore if the feveral fractions be added together,
and reduced into an equivalent decimal, we fhall have the
Logarithm of the given number; but becauſe to multiply by
and divide by 9 is the fame thing, if the firft fraction be
reduced into a decimal, and that divided by 9, and each
fucceffive quotient by the fame, we ſhall have a ſeries of
quotients, which being divided by the feveral co-efficients
1, 3, 5, 7, 9, &c. the fum of theſe laſt quotients will be the
Logarithm of the number given; and in order to obtain
the Logarithm true to any number of places, it will be
neceffary to continue the fraction to one or two more
places, than the intended number of places the Logarithm
is to confift of; ſo that to have the Logarithm true to
10 places, it will be convenient to find the value of in
decimals to 11 or 12 places, as in the following operation.
11/==
An APPENDIX.
363
333.333. 333. 333
-37. 037. 037. 037 its
I
4. 115. 226. 337 its
333.333.333.333
12. 345.679. 012 +
823.045. 268
-65. 321. 053
5.645.0294-
513. 185-
48. 248
457. 247. 371 its
50. 805. 263 its
5.645. 029 its
627. 225 its
69. 691 its
7. 743 its
860 its
す
​73
75
I
J
Ty
96 its t
I
ΣΤ
4. 646
455
·45 +
5-
Whence the natural Log. of 2 is 346.573.590. 280
And the Neper's Log. of 2 is
2
-693. 147. 180. 560
Hence if the Neper's Log. of 2 be doubled will give
1386. 294. 361. 120, for the Neper's Log. of 4, and this
again being doubled will give 2772. 588. 722. 240, for the
Neper's Log. of 16, &c. Or if the Log. of 2 be trebled,
or multiplied by 3, we ſhall have 2079. 441. 541. 680, for
the Log. of 8, &c.
And, if the Logarithm of 8 be multiplied by the Log.
of 1, we fhall have the Log. of 10; and to find the Log.
11 we have a =1,
, whence x=
z 2, and d = 4,
and confequently
1, b
I
d
I
d2
I
and
From whence
Z
≈2
81
the Log. of 10 will be found as follows.
ET
R
&
I
8T
III. III. III. III
1. 371. 742. 112 its
16.935. 087 its
209. 075 its
2. 851 its
32 its
The natural Log. of 1 is
1
And the Neper's Log. of 1 is
III. III. III. III.
457.247.371
3. 387.017
29.868
287
3
111. 571. 775. 657
2
223. 143. 551. 3141
The Log. left found being added to the Log. of 8 found
before, we ſhall have 2302. 585. c92. 994, for the Neper's
Log. of 10.
But as neither the natural Logarithms, nor thoſe of Neper,
the celebrated inventor of Logarithms, are fo proper for
calculation as could be wished, the Lord Neper himſelf,
with the affiftance of Mr. Henry Briggs, altered their form,
and made the Logarithm of 10 to be 1,0000000000, &c.
and
364
An APPENDIX.
and not 2.30258509299, &c. that is, he made the number
10 the 100000000ooth term in the feries, whence the Lo-
garithm of 100 will be 20000000000, &c. and the Loga-
rithm of 1000, will be 30000000000, &c. whence the
Logarithms between 1 and 10 will begin from o, that is,
they will have o for the first term towards the left-hand,
becauſe they are each of them leſs than the Logarithm of
10, which has unity or 1 in the firſt place; in like manner
the Logarithms of all numbers betwixt 100 and 1000 begin
from 2, becauſe they are greater than the Logarithm of
100, which is fixed at 2, and less than the Logarithm of
1000, which is made equal to 3, &c.
The firft figure of every Logarithm towards the left-hand,
which is feperated from the reft by a point, is called the
index of that Logarithm; becauſe it points out the higheſt
or remoteſt place of that number from the place of unity in
the infinite ſcale of proportionals towards the left-hand.
Thus if the index of the Logarithm be one, it ſhews that
its higheſt place towards the left-hand is the tenth place
from unity; and therefore all Logarithms which have 1 for
their index will be found between the tenth and hundredth
place in the order of numbers. And for the fame reafon
all Logarithms which have 2 for their index will be found
between the hundredth and thoufandth place in the order of
numbers, &c. whence the index or characteriſtic of any
Logarithm is always lefs by one than the number of figures
in whole numbers which answer to the given Logarithm.
As all ſyſtems of Logarithms whatever are compoſed of
fimilar quantities, it will be eafy to form, from any ſyſtem
of Logarithms another fyftem in any given ratio, and con-
ſequently to reduce one table of Logarithms into another of
any given form. As for inftance, in the fyftem given let
a, b, and c be the Logarithms of the three given numbers
A, B, and C, whereof the third is equal to the product of
the other two multiplied together. Then will a+b=c by
the definition we have given of Logarithms. Let us now
imagine all the Logarithms of this given fyftem to be dou-
bled, then will a, b, and c be changed into 2a, 2b, 2c;
but as a+b was equal to c in the former fyftem, ſo in the
latter za 26 will be equal to 2c. Whence it follows that
we may reduce one fyftem of Logarithms into another of
any given form by faying, as any one Logarithm in the
given form is to its correfpondent Logarithm in the new
form; fo is any other Logarithm in the given form, to its
correfpondent Logarithm in the required form; and hence we
are taught how to reduce either Neper's or the natural Loga,
rithms into the form of Brigg's, and the contrary.
T
For
An APPENDIX.
365
For as 2. 302. 585. 092. 994, the Neper's Logarithm of
10, to 1.0000000000, the Brigg's Logarithm of 10; fo is
any other Logarithm in Neper's form, to the correfpondent
tabular Logarithm in Brigg's form: And becauſe the two
firſt numbers conſtantly remain the fame; if the Neper's Lo-
garithm of any one number be divided by 2.302.585, &c.
or multiplied by 434. 294. 481. 903, the ratio of 1.00000,
&c. to 2. 302. 58, c. as is found by dividing 1.000000,
&c.
c. by 2. 302. 58, &c. the quotient in the former, and the
product in the latter will give the correfpondent Logarithm
in Brigg's form, and the contrary. And after the fame
manner the ratio of natural Logarithms to that of Brigg's
will be found 868.588. 963. 806.
If inſtead of finding a decimal fraction equal to
d
Z
we
divide the reciprocal ratio of the Neper's or the natural Loga-
rithms, to that of Brigg's by the reciprocal ratio of d to x,
and divide that quotient by the fquare of x, we ſhall have
a feries of quotients, which being divided refpectively by
the feveral co-efficients 1, 3, 5, 7, 9, &c. will produce a
new ſeries of quotients, whofe fum will give the Brigg's
Logarithm of the number propoſed.
Let it therefore be required to find the Brigg's Logarithm
of 2 to ten places, from the reciprocal ratio of the natural
to the Brigg's Logarithm.
Put R-868. 588.963. 806, and becauſe a1, b—2, d—1,
I
and ≈≈3; therefore R x÷+
X-+
I
I
I
I
1
I
X +
X
5
243
7^21879
3 3 27
&c. the Logarithm of 2.
R289. 529. 654. 60
焉
​32. 169. 961. 62 its
3. 574. 440. 18 its
397. 160, oz its
44. 128. 89 its
=289. 529. 654. 60
II II
10. 723. 320. 54
714.888.04
56. 737. 14
4.903. 21
445.75
4.903. 21 its
544.80 its
T
41.91
60, 53 its
IS
4.03
6. 72 2 its
74 its T
40
4
The Log. of 2 in Brigg's form is
301. 029. 995. 66
Again, let it be required to find Brigg's Logarithm of 3.
ď
I
Becauſe a-1, b—3, d—2, and ≈≈4; therefore
x
2
And
366
An APPENDIX.
And
R
2
+
3
R
X +
8
I
X
R,
c. will be the Loga-
5 32
rithm of 3; wherefore, if R be divided by 4, and each
quotient fucceffively by the fame number, and thofe again by
the indices of the odd powers, the fum of theſe laft quotients
will be the Brigg's Logarithm of 3; but becauſe this ſeries
converges very flow, the fame Logarithm of 3 may be found
much quicker, by finding the difference between the Loga-
rithm of 2 already known, and the Logarithm of 3 required.
Put therefore 2, and 6-3, then will z5, and d=},
d²
I
and - the common multiplicator equal to › wherefore
༤
25
if R be divided by 5, and that quotient by 25, &c. and
thefe feveral quotes by 1, 3, 5, 7, &c. the fum of theſe
laft quotients added to the Logarithm of 2, will give the
Logarithm of 3.
Operation,
÷ R=173.717. 792. 76
6. 948. 711. 71 its
27
25
25
25
23
277.948. 47 its
11. 117. 94 its
444. 73 its
3
5
TI
17.79 its
71 its
T 3
— 173. 717. 792. 76
Diff. between the Log. of 2 and 3 =
Wherefore if to the Logarithm of 2 =
2. 316. 237. 24
55.589 69
1. 588. 28
49.41
1.62
176. 091. 259. 05
301. 029. 955. 66
the difference juft found, be added, we fhall have the Loga-
rithm of 3477. 123. 254. 71.
12
Z
Again let it be required to find the Log. of 7.
Put a=6, b = 7, then will z=13, and d—1, whence
I
Z
, and therefore x R
169
confequently.
I
169
T39
66.814. 535.68
d
395· 352. 28 its
3
2. 339. 36 its
13. 84 its
8 its
66. 814. 535. 68,
I
787
66.814. 535.68
131. 784.09
476.87
1. 98
I
The diff. of the Log. 6 and 766.946. 789. 63
To the Log. of 6
Add the diff. found
The fum is the Log. of 7
Wherefore
778. 151, 250. 38
66.946. 789.63
845. 098.040. of
The
An APPENDIX.
367
=
The Logarithm of all compofite numbers, that is, all fuch
as arife from the multiplication of others called factors, are
eafily found by the addition of the Logarithms of thofe
factors; Thus the Log. of 4= Log. 2 - Log. 2, and the
Log. to Log. 2+ Log. 5, and therefore the Log. of
5 Log. 10- Log. 2; the Log. 6 — Log, z + Log. 3;
Log. 8 = Log. 2 Log. 4; and the Log. 9 = Log. 3 +
Log. 3. Therefore from the Logarithms of the prime
numbers 2, 3, and 7, which we have already found, all
the Logarithms of the numbers under 11, may be compu-
ted by addition and fubtraction only. And after the fame
manner may the whole cannon be conftructed, the prime
numbers being eafily computed by the Theorems we have
laid down.
The excellence of this method will be the more confpicu-
ous if we compare the foregoing computations of the prime
numbers with each other; for then it will appear, that as
the prime number itſelf increaſes, the operation decreaſes,
and the fewer fteps are requifite; till at last the firſt ſtep
will be fufficient to find the difference to any number of
places requifite.
Thus in the common tables, which extend but to feven
places in decimals, when the difference d becomes equal to
the one two hundredth part of the fum z, the firſt ſtep,
d
Z
x R, will give the difference between the Logarithms
true to the fame number of places.
For if 868. 588. 9 be divided by 201, the fum of the
numbers 100 and 101, the quotient 0043213, will give the
difference between the Log. of 100 and 101, true to 7 places,
which therefore being added to 2.0000000, will give 2.00-
43213, for the Logarithm of 101; fo that the Logarithms
of the prime numbers under 100 being found, the Loga-
rithms of all the prime numbers above 100 may be found
by one fingle divifion, and the Logarithms of any intermedi-
ate numbers by addition and ſubtraction only.
As the Logarithms 1, 10, 100, 1000, 10000, &c, in
Brigg's form are 0, 1, 2, 3, 4, c. it is eaſy to conceive
that the Logarithms of all numbers which increaſe or de-
creaſe in a tenfold proportion, differ from each other only
in their indices or charactereftics, the fractional number
remaining conſtantly the fame: And hence it is that the
fractional number .30102999566, which is the Logarithm of
2, when the index is a cypher, becomes the Logarithm of
20 when the index is 1, of 200 when it is 2, of 2000,
'
when
#
368
An APPENDIX.
when it is 3, &c. Alfo the fame Logarithm when its index
2
is -I becomes the Logarithm of when the index is
2
of
when -3 of
100
2
1000
ΙΟ
2
&c. the fractional number ſtill
remaining the fame; for becauſe as 1 : 2
1: 2: 10: 20:
100: 200 :: 1000: 2000, &c. the diſtance between 1 and
2, 10 and 20, 100 and 1000, 1000 and 2000, &c. are
equal to each other, and the number of terms in the infi-
nite fcale of proportionals between the numbers 1 and 2,
10 and 20, 100 and 200, 1000 and 200o, that is, the
Logarithm of that diſtance will be the fame; and fince the
Logarithm of 1 is 00000000000 of 10 is 1.0000000000, of
100, 2.0000000000, and the Logarithm of 2 is.30102999566,
the Logarithm of 20 will be 1.30102999566, of 200,
2.30102999566, &c.
And becauſe the numbers 37251, 3725.1, 372.51, 37.251,
3.7251, .37251, .037251, .0037251, &c. are continual
proportionals in the ratio of 10 to 1, the number of terms
between each in the infinite ſcale of proportionals will be
equal and therefore fince the Logarithm of 371251, is
4. 5711379358, the other numbers will be 3.5711379358,
2.5711379358, 1.5711379358, 0.5711379358, -1.5711-
379358, —2.5711379358, -3.5711379358, &c.
This property of having the fame number for the Loga-
rithms of an infinite feries of numbers by increafing or
diminiſhng the characteristic by unity is peculial to Brigg's
Logarithms, which renders them preferable to any other
yet invented.
It has been already fhewn, that
12
1+x- I
I
I
I
- X x-
x²+
72
2
3
x4, &c. equal to I will be the Logarithm of
the ratio of 1 to 1x, where n repreſents any infinite index,
any number taken at pleaſure. Now if L be put
and
for the Logarithm itſelf, then will
12
J
72
+x-1=L, confequent-
-72
ly, +x will be equal to L+1, and 1+x=L+i;
n
72
nn- -n 223 +372² + 2x2
2
and becauſe L+ 1 =1+¬L+—Ľ²+
I³ of
6n³ + 11m² -бn
24
2
6
L*, &c. (by_the_general
Theorem
An APPENDIX.
369
Theorem in page 360) when is finite, when n becomes
n
n
I
infinte L+ will be equal to 1+L+n² L²+,
I
I
2
I
π³ L³ +
24
nª L++ — ½³ L³ +
Ls
no
—— 1° Ls +
24
120
720
5040
I
1
n'
Xx
n
»¹ L², &c. =1+x. Alſo becauſe xx++
I
I
I
x²+-x4 +
3
5
I
x³ + — x• &c. — 1 — ï—qå is the
I
Logarithm of the ratio of 1 to 1-x, therefore I - 1
I x
72
WIN
➡L, and confequently 1 -x
= I
L, and i x
-n
1 —L, wherefore by the fame theorem 1—x—ı~L, will
1
I
be equal to In L+ 2 n² L² — ♂ n³ L³ + 24 xª L^—
I
I
ns Ls + no Lo
120
+
720
5040
6
I
n² L², &c.
222
n if-
2
24
Whence 1+x=1±L+=”²L²±÷”³L³+
L4+25 Ls, &c. will be a general Theorem for
120
finding the number from the Logarithm given, of any form
whatſoever.
And becauſe — 1000000, &c. in Neper's form, 1+*
1
I
2
2
6
L³ + L4
+1
I 20
24
Ls, &c. will
give the number anfwering to any Logarithm in Neper's form.
Hence any one term of the ratio, whereof L is the Loga-
rithm being given, the other term will be readily found;
for by putting a for the leffer of the two terms, and b for
I
2
+++
the greater, a×1+L+ — L²+ — L³+/
2
24
L³, &c. will give the greater, and 6×1-L+L²
I
L³ + - L4
24
form.
120
2
I
120
Ls, &c. will give the leffer, in Neper's
:
A a
Ji
.370
An APPENDIX.
I
ax
a x 1 + n Lit
if
Or,
I
223
1
π² L² - + - - ½³ L³ +
24 L4 +
2
24
I
I
n³ L³, &c. =b: and a XI-n L +
n² L²
120
2
6
I
I
n³ L³ + 24 L4
d
24
120
n³ L³, &c. — a in any form.
Let the difference between the given Logarithm and
the next neareſt tabular Logarithm; then will axi+d+
d³, &c.— N. Or,
I
+ =/
I
I
24-+-
2
6
24
120
I
I
I
I
bxi-d+
ď²
d³ +
d+
2
6
24
N, the number required in Neper's form.
1
72
120
d³, &c. =
Or a x1+nd +
n³ d³, &c. — N.
I
I
I
n² d² +
n³ d³ + 12+ d4af
2
6
24
I 20
I
I
And 6 × 1 —nd +
n² dz
π³ d³ +
n4 d4
2
6
24
I n³ d³, &c. N, the correfpondent number in Loga-
120
=
rithms of any fpecies.
7
Let it, for example, be required to find the number an-
fwering to 7.5713740280 in Brigg's form.
Becauſe 7.5713710453, the neareſt tabular number to this
is lefs than the given number, put 37271000, the number
anſwering to it a, and oocoo29827, the difference be-
tween the two Logarithms =d, then will π — 230258, &c.
multiplied by 37271000, the next neareſt number in the table,
give ocooo68683; which being increafed by 1, and multi-
plied by 37271ooo, the next neareft number in the table,
will give 37271255.988 for the number anfwering to the
given Logarithm: Or, if ooo0068683 be multiplied by
37271000a, the product 255.998, added to a — 37271000
will give 37271255.988 for the number required, the fame
as before.
i
Hence, and from the general Theorem, may the number
anſwering to any Logarithm be found, and conſequently an
anti-logarithmic canon be conftructed, fhewing the natural
numbers anfwering to every Logarithm; whence the num-
bers anfwering to any Logarithm might be found with. the..
fame eaſe as we find the Logarithm of any number in the
common Logarithmic Cannon. Such a Cannon has been
lately published by the ingenious Mr. Dodfon, which has
rendered the Logarithmic Tables compleat.
Having
An APPENDIX.
371
Having thus fhewn the method of conftructing the Loga-
rithmic Tables, we ſhall now proceed to fhew how the Lo-
garithmic fines, tangents and fecants are computed; but it
will be requifite previouſly to explain the manner how the
natural fine, tangent, fecant, &c. of any arch may be found.
In order to which let C (Plate IX. Fig. 19) be the center
of a circle, CA CS- the radius, ASa any arch
thereof, Srs its right fine; then will Cr, the co-fine
√ I ss, by the 47.1.
Let CM be another radius of the circle, infinitely near to
CS; then will Mns be the fluxion of the fine Sr, and
the infinitely ſmall arch MS=a, the fluxion of the arch
AS. And becauſe the triangles CS r and M » S are fimilar,
it will be as Cr: CS :: M n : SM; that is √ I—ss : 1 :
s: a; wherefore a a-
I
SS
n
that is, if the fluxion of
the right-fine be divided by the fquare root of 1-ss, the
intregal, or flowing quantity of the quotient will be the
arch itſelf.
The fquare root of 1-ss, being extracted will be 1
I
52
2
8
I
s, &c. by which the fluxions muſt be
16
divided, which will give the fluxion of the arch SA, as follows,
Ι
2
I
16
I
$4
so, &c.
>
وک
ម
5
6
S
2
16
I
I
I
ܨ
—ss, & co
2
8
16
I
I
+
2
8
-
8
1 +31:00 0/00
4
2
+
+
I
16
&c.
sos, Et:
I
stst
8
$,&c.
3
st;
3
s, Eco
16
5
A 22
+ sos, &ci
16
Confequently
372
An APPENDIX.
Confequently st
2
5
3
stst
3 + 1/6 56 s, &c. = a,
8
the fluxion of the arch, whofe fluent or flowing quantity is
I
6
sufe s3 of
3
40
5
55 of
+ 57, &c.
Therefore a st
IX I
st
53 m
2x3
112
3
53 +
+
6
40
I
+
1 X 1 X 3 X 3
5
112
s", &c. or a
`s", &c.
I X1 X3 X3 X5 × 5
es f
-2 X3 X4 X5 2X3X4X5×6×7
Hence we may find the length of any arch of a circle, the
right-fine of that arch being given.
And becauſe the radius or femidiameter of a circle is
equal to the fide of a hexagon infcribed in a circle (Cor. 1.
Prop. 15. Book 4.) or chord of 60 degrees, therefore the fine
of 30 degrees will be equal to half the radius: And confe-
quently when the radius of a circle is given, the fine of 30
degrees is given alfo.
1.
Let it therefore be required to find the length of the arch
of thirty degrees, to ten decimal places, which is abundant-
ly fufficient for any common purpoſe, the radius being
Becauſe the radius 1, the fine of 30 deg.
=s, will be
, and confequently its fquare, its cube, &. And
as to multiply by, and divide by 4, is the fame thing, it
follows that if the right-fine .5 be divided by 4,
and the feveral quotients refulting from thence be multiplied
1
by 1, 3, &c. the fum of theſe products will be the
40
length of the arch required.
S =.5000, 0000, 000 —
1250, 0000, 000 its
$3
5
ک
J
II
13
IS
I 9
521
$23
525
527
312, 5000, 000 its
78, 1250,000 its
19, 5312, 500 its
4, 8828, 125 its
I, 2207, 031 its
3851, 758 its
762, 939 its
190, 735 its
47, 684 its
11, 921 its
2, 980 its
ΙΤΣ
3 S
T15
୪
2878
231
T3312
143
6+35 &C.
570389
1215 &C.
J27378,
550302,
4519 &C.
6799 &C.
928314
3501 &C.
357108,
745 its 2244 &C.
The length of the arch of 30 deg.
•5000, 0000, 000
208, 3333, 333 +
23, 4375,000
3, 4877, 232 +
5933, 974
1092, 391
211, 826
42, 617 te
-8,813 +
1, 863
400
87
19
-4
·4 +
=.5235, 9877, 559
If
{
An APPENDIX.
373
If this arch be multiplied by 6, we ſhall have the length
of the arch of the femicircle when the radius is 1, or the
length of the whole circumference when the diameter is 1.
•5235, 9877, 559
6
3. 1415, 9265, 354
The abfolute neceffity of finding, as near as poffible,
the proportion between the diameter and circumference of
a circle, for exactly true it never can be found, has engaged
fome of the greateſt Mathematicians to approximate as
near as poffible to the truth. Van Ceulen carried it to 32
decimal places, which he ordered to be engraved on his
Tomb-ſtone, as a memorial of ſo great a work. His num-
bers are as follows,
If the diameter be 1.0000000, &c. the circumference
will be 3.141592,653589,793238,462643,383279,50288+.
Since Van Ceulen, the indefatigable Mr. Abraham Sharp,
has carried it to twice the number of places, and fhewn
us that if the diameter be 1.0000000, &c. the circum-
ference will be 3.141592,653589,793238,462643,383279,
502884, 197169, 399375, 105820,974944, 592307,816405,
2+.
And the learned Mr. John Machin, has carried it to 100
places, which is a degree of exactnefs far exceeding all
imagination. And is as follows.
If the diameter of a circle be 1,000, &c. the circumference
will be 3.141592,653589,793238,462643,383279,502884,
197169, 399375, 105820,974944,592307,816405, 286208,
998628,034825,342117,0679+of the fame parts.
Becauſe a=s+ -—-s³ +
3
sit
s7, &c. Therefore,
40
II
I
3 = a
a³ +
as
a", &c.
I 20
5040
Or,
I
I
I
a³ +
as
a", &c.
2x3
Aa
2X3X4X5 2X3X4X5X6
For puts A a + Ba³ + Ca³, &c,
I
I
+ / A³ a³ + =—=— A²
A Ba", &c.
6
I
Then
6
$5
A a 3
And 3
2
+ 3 As as, &c.
a
Confequently
40
40
374
An APPENDIX.
Confequently A a—a, and A = 1; alſo B+
I
o, and B =
6.
I
I
Alfo C+ AB+ A³
3
A3
6
3
2
40
I
I
40
2
6
3
I
+ wherefore A1, B
;
40
6
c=+
120
&c.
c. wherefore sa-
o. And C-AB-AS-×--
I
120
2
6
I
a³ + a³, &c. Q. E. D.
120
Let it be required to find the natural fine of 10 degrees.
Becauſe the length of the femi-circumference is 3.1415,
I
9265, 354, therefore the length of 10 deg. or will be
.1745, 3292, 5196.
18
Put therefore a —
Then will
3
a
And a-
½ a
a³
Again+Ta' =
And a -
I
as
ΤΣ
Alfo
And aa 3
+
a5
720
I
5040
I
3045
to a 7
a7=
1745, 3292, 519
8, 8609, 615
1736, 4682, 904
134, 960
1736, 4817, 864
- 97
1736, 4817,767
Therefore s.1736, 4817,767, the fign of 10 degrees
required.
The fine of an arch being given, its co-fine may be eaſy
found; for becaufe CSqSRq + CRq (by 47.1) fee Plate
IX. Fig. 3. therefore CSq SRq CR. That is, if
from the fquare of the radius, be taken the fquare of the
right fine, the ſquare root of the remainder will be the co-
fine required.
Therefore if from the fquare of radius 1, be taken the
fquare root of the right-fine
½ a³ + ½ a³, &c.
a
I
the fquare root of the remainder will be the co-fine required.
Thus,
An APPENDIX.
375
Thus,
S
S
a
2
୯
½ a³ + ½ as, &c.
T20
½ a³µ‚½o a³, &c.
720
} a+ + + ½ o a³, &c.
— — a++ zaº, &c.
Σ
6
+io a³, &c.
T20
}; a + + + 3 a°, &c. which taken from the
=1, leaves I
55— a²
3
2
कड़
fquare of radius
· a² + =/; a+
2
45
a“, &c.
the fquare root of which will be the co-fine required.
1
aa
+ & a+
I
2 — 1 aa)
aa+
2
写
​+ } aº, &c. ( 1 — ½ a ² † z
• a² + = a +
a
120 46, &c.
aa + = a +
aa + // a4
2—aa, &c.) + ½ at
2 -
૨૬
+
TÉ 04
aa, &c.)
—
Wherefore the co-fine I
Or the co-fine — 1 —
I
I
43 a^, &c.
I
27
a
26, &c.
3έo a6, &c.
360
I
3ko a³, &€.
1
1/2 a² + 2 = a +
I X 2
I
20
‚½% a³, &c.
I
a²
+
a+
1 × 2 × 3 × 4
a6, &c.
1 X 2 X 3 X 4 × 5 × 6
The fines being thus obtained, the tangents and fecants
are eaſily found as follows.
For becauſe the triangles CRS, CAT, (Plate IX. Fig. 3.)
are fimilar, it will be,
As CR, the co-fine, is to RS, the right-fine; fo is CA,
the radius, to AT, the tangent.
And becauſe the radius =1, therefore the quotient of
the right-fine divided by the co-fine will give the tangent of
the fame arch.
Hence the tangent feries are eaſily found; for if a -
I
a3
6
I
1
I
+ as, &c. the right-fine, be divided by 1 - aa
120
2
+ at, &c. the co-fine, the quotient will give the tangent
24
required, as in the following example.
A a 4
376
An APPENDIX.
//
1-‡aa+za+, &c.) a— a³ + a³, &c, (a+{a³+; } }
a
120
a² + ½ ½ a³, &c.
(as, &c.
the
Hence if a
tangent will be a +
1382 I I
7514 a¹¹, &c.
37923
a³ + zja³,
&c,
3
I
3 as, &c.
2
+3 as, &c.
length of any arch, the correſpondent
as
a³ + 3 a³ +33a² + 288 3 a +
2
I
Again, becauſe the triangles
I 7
37
62
2835
ACT, GBC, are fimilar, it
will be, as AT, the tangent, to AC, radius; fo is CB, ra-
dius, to BG, the co-tangent.
Hence the radius is a mean proportional between the tan-
gent and co-tangent of an arch, and becauſe the radius —1
if I be divided by a + a³ + as, &c. the tangent, the
quotient
I
I
a
3
I
2
I
2
- a³
as
45 945
2
I
·a"
4725
93555
a, &c. will be the feries for finding the co-tangent, from
the arch firſt given.
Again, becauſe the triangles CRS, CAT, are fimilar, it
will be as CR, the co-fine, to CS, the radius; fo is CA,
the radius, to CT, the fecant.
Hence the radius is a mean proportional between the fe-
cant and co-fine of an arch.
༡
And therefore if I be divided by 1-
277
+
Σ
a4
5 if
1
‚½ aº, &c. the co-fine, the quotient + a + at t
Z z a ° + z 2 2 7 a³ +3758835 ato, &c. will be the ſeries
for finding the fecant from the arch firſt given.
ફ્
I
720
50521
2880
A cannon of natural fines, tangents, and fecants being
conſtructed according to the foregoing examples, it will be
eafy to find the correſponding artificial fines, tangents, and
fecants; for the Logarithms of the former will be the latter.
But the Logarithmic fine, tangent, or fecant of any arch
may be found, independantly of the tables of Logarithms;
for by putting s for the fine of any arch, and q
I
fhall have
x q
12
3
1
5
7
I
we
9+9³ + 9² + ½ q², &c. for the Lo-
garithmic fine of the fame arch, and if inſtead of s, in the
former feries, we put for the tangent, or c for the fecant
of any arch, the fame feries will produce the Logarithmic
tangent or fecant of the given arch.
Or if inftead of s, the right-fine, in the above feries we
n 플
​q ³ — 2114 9° — #443'a¹°, &c. be the
put a, for the length of the arch itſelf then will × -
a) — ; et — =
6.
2529
{
a
8
I 1º,
T4775.
Logarithmic
An APPENDIX
377
Logarithmic co-fine of the fame arch, in Brigg's form, the
radius being 10.0000000000, and n = 2. 3025,8509, 29.
We have already fhewn that if the radius of a circle be
put equal to 1, the length of the arch of 10 degrees will be
equal to 1745, 3292, 519, of the fame parts; put this there-
fore equal to a.
Then will 2
And
ΤΣ
24
Alfo
And
4
5
8
1 7
2320
6
7
8
0152, 3087, 10
7732,65
62, 81
58
Therefore aż a+++} a²+zšï a³ 0153,0883,14
252
'This value being taken from n = 2. 3025, 8509, 29, will
leave 2, 2872, 7625, 15, which being multiplied by 4342,
9448, 190, will give the fractional part of the Logarithmic
fine of 80 degrees.9933, 5145, 89, to which prefixing 9
for an index, becauſe the radius is equal to 10, will give
9.9933, 5145, 89, for the Logarithmic fine of 80 degrees in
Brigg's form; and after the fame manner may the fine of
any arch be found independant of the Logarithmic tables,
and confequently the whole table of fines conſtructed.
Having thus computed a table of Logarithmic fines, the
Logarithmic tangents and fecants may be had by common
addition and ſubtraction only. For,
1. As the co-fine of an arch, is its right fine; fo is the
radius, to the tangent of the fame arch.
Therefore if to the right-fine of an arch we add the
radius, and from that fum fubtract the co-fine of the fame
arch, the remainder will be the tangent required.
Thus, To the Log. fine of So degrees 9.99335,14589
Add the radius
And fubtract the co-fine of So degrees
=
10. 0000000000
19.99335,14589
9.23967,02300
The remainder is the Log. tang. of 80 deg.10.75368,12289
2. As the co-fine, is to radius; fo is radius to the fecant
of the fame arch.
Therefore, from twice radius fubtract the co-fine, and the
remainder will give the fecant of the fame arch.
Thus, from twice radius
Take the co-fine of 10 degrees
The remainder is the fecant of 10 deg.
20.0000000000
9.99335,14589
10.00664,85411
Hence it appears that if the Logarithmic fine and co-fine
of an arch be known, its tangent and fecant may be rea-
dily found; and confequently the whole cannon eafily
computed.
'The
J
378
An APPENDIX.
The ingenious Mr. Sympfon, in his Trigonometry, page
47 & feq. has given us a very compendious method of
finding the Logarithms of large numbers, one from another,
by addition and ſubtraction only, which is as follows.
1. Let A, B, C, denote any three numbers in arithmetical
progreffion, not less than 10000 each, whereof the common
difference is 100.
2. From twice the Logarithm of B, fubtract the fum of
the Logarithms of A and C, and let the remainder be di-
vided by 10000.
I
3. Multiply the quotient by 49.5, and to the product add
1 part of the difference of the Logarithms of A and B;
then the fum will be the excefs of the Logarithm of A + 1
above that of A.
4. From this excefs let the quotient (found by Rule 2.)
be continually fubtracted, that is, firft from the exceſs itſelf,
then from the remainder, then from the next remainder,
&c. &c.
5. To the Logarithm of A add the ſaid exceſs, and to the
fum add the first of the remainders; to the laſt ſum add the
next remainder, &c. &c. then the feveral fums, thus arifing,
will exhibit the Logarithms of A1, A2, A+ 3,
&c. respectively.
Thus let it be propoſed to find the Logarithms of all
the whole numbers between 17900 and 18100; thoſe of
the two extremes 17900 and 18100, and that of the mean
(18000) being given.
Then the Loga- ΓΑΙ
rithm of
we ſhall have
B being equal to
CJ
2 Log. B-Log. A-Log. C
10000
4.252853031
4.255272505
4.257678575
=.00000000134
(fee Rule 2.) which multiplied by 49.5, and the product
Log. B-Log. A
added to
100
gives .00002426107 for the
excess of the Logarithm of A+1 above that of A (by
Rule 3.) From whence the work, being continued accord-
ing to Rule 4 and 5, will ftand as follows.
,00024
An APPENDIX.
379
000024 25107 excefs
134
25973 ft Rem.
134
25839 2d Rem.
134
25705 3d Rem,
134
25571 4th Rem.
134
25437 5th Rem.
134
25303 6th Rem.
134
25169 7th Rem.
134
25035 8th Rem.
134
24901 9th Rem.
134
24767 10th Rem.
&c. &c.
4.25 2853031 Log. of 17900.
2426107 excefs
287729207 Log. of 17901.
2425973
290155280 Log. of 17902.
2425839
292581019 Log. of 17903.
2425705
295006724 Log.of 17904.
2425571
297432295 Log. of 17905.
2425437
299857732 Log, of 17906.
2425303
302283035 Log. of 17907.
2425169
304708204 Log. of 17908.
2425035
307133239
2424901
309558140 Log. of 17909.
&c.
&c.
Note, The Longarithms found by this method, in num-
bers between 10000 and 20000, are true to 8 or 9 places of
figures. Thofe of numbers between 20000 and 50000 err
only in the 9th or 10th place; and thofe above 50000
are true to ten places at leaſt.
Having fufficiently explained the manner of conftru&t-
ing the tables of Logarithms, it remains that we fhew the
uſe of thoſe tables.
1. We having already obferved that addition of Log
arithms anfwers to multiplication of natural numbers, and
fubftraction to divifion. Thus if it were required to
multiply 8.5 by 10, the operation will be as follows,
To the Log. of 8.5.
Add the Log. of 10
The Sum
=0.9294189
1.0000000
1.9294189
Which is the Log. of the product required, and the num-
ber corresponding thereto is 85.
Or if it were required to divide 9712.8 by 456, it will
ftand thus,
From
380
An APPENDIX.
From the Log. of 9712.8
Take the Log. of 456
Remains
= 3.9873444
2.6589648
1.3283796
Which is the Log. of the quotient required, and whofe
correſponding number is 21.3.
To find the Complement of a Logarithm.
Begin at the left-hand, and fet down the difference be-
tween each figure and 9, except the firſt, which muſt
be fubtracted from ten. Thus you will find the com-
plement of the Log. of 456, viz. of 2.6589648 to be
7.3410352.
2. To perform the rule of Proportion by the Logarithms.
Add the Logarithms of the fecond and third terms to-
gether, and fubtract the Log. of the firſt from that ſum ;
the remainder will be the Log. of the fourth term or
number fought. Or inftead of fubtracting the Log. of
the first term, add its complement, and the refult will
be the fame: But obferve that in adding a complement
you muſt always cancel 1 in the tens place of the index,
for every complement fo added.
Thus, let it be required to find a fourth number in
geometrical proportion to the three following, viz. 4, 9, 12.
To the Log. of 9-
Add the Log. of 12
And from their fum
=0.9542425
1.0791812
2.0334237
0.6020600
=1.4313637.
Take the Log. of 4
The remainder is the Log. of 27
So that the fourth number required is 27.
Or thus,
Complement of the Log, of 4 is -
Log. of
Log. of
Sum is Log. of 27
9.3979400
9-
0.9542425
12
1.0791812
-1.4313637
The fame as before, except in the index, where I in the
place of tens muſt be cancelled, as we before obſerved.
Example 2. Suppofe 100l. in one year, or 365 days, gain
67. intereft; what will 51737. gain in 321 days?
Anfwer, 272.96431. or 272%. 195. 3d.
nearly.
Comp.
Fig.3.
Plate IX. Facing Pag.380.
Fig.1. E
Fig.
2. A
B
B
H
B
A
Fig.4.
E
Fig.5.
T
C
R
D
E
P
H
B
10
A
B
D H
Fig. 6.
Fig.9.
N
T
Fig./
Y
B
୯
E
D
H
R
P
E
A-
-B
Z
Y
Y--B
Z
Fig.13.
Fig.10.
A
Z
Fig.u.
2
BY
Z
N
Z
2
B
E
R
Fig. 8.
W
M
F
Fig.14.
Z
C
D...
Y
E
E
Y
Z
A
E
Fig.16.
B
b
Fig. 15.
B
d Fig. 27. A
M
Fig. 12.
T
Fig.18.
n
m
B
X
A
B
M Fig.19.
n
R
An APPENDIX.
381
Comp. of the Log. of ico
8.0000000
Comp, of the
Log. of 365
7.4377071
Log. of
6
0.7781513
Log. of
5173
3.7137425
Log. of
321
2.5065050
Log. of
-272.9643
72.4361059
In the above operation, 2 in the place of tens in the
index is cancelled, becauſe two complements are added.
3. To raiſe Powers by Logarithms.
Multiply the Logarithm of the given number by the
index of the required power, the product will be the
Logarithm of the power fought.
Example. Let it be required to find the cube of 32 by
the Logarithms.
Which multiply by the index of the power
The Log. of 32 is
is-
The Log. of 32768
Which is the cube of 32 required.
1.5051500
3
4.5154500
Example 2. Let it be required to find the cube of 0.009
by the Logarithms.
The Log. of 0.009 is (fee page 368)
Which multiply by
Which is the Log. of 0.000000729
3.9542425
3
-7.8627275
N.B. When the Log. has a negative index, as in the
above example, whaterever is carried from the decimal part
of the Log. muft be confidered as affirmative, and fub-
tracted from the multiplication of the negative index; for
the decimal part of every Log. is affirmative, whether the
whole Log, taken together be fo or not. Thus in the above
example, the negative index 3 being multiplied by 3, pro-
duces 9, from which the 2 carried from the decimal part
being fubtracted, gives 7 for the index of the Log.
required.
Example 3. Let the 6th power of 0.0032 be required.
The Log. of 0.0032 is
Which multiplied by
The product is
6
3.50515500
Which is the Log. of -
15.03093000
0.0000000000000010739
4. To
382
An APPENDIX.
4. To extract the Roots of Powers by the Logarithms.
Divide the Log. of the number by the index of the
power, the quotient is the index of the root fought.
Example, Let it be required to extract the cube root
of 6751269.
The Log. of 6751269 is 6.8293854, which being divided
by 3, the index of the power, gives 2.2764618. the Log. of
189, the cube root fought.
Example 2. Let it be required to extract the root of the
6th power of 0.00000,00000,00000,00000,00000,00000,63
by the Logarithms.
The Log. of 0,00000,00000,00000,00000,00000,00000,
63 is 31.8061800, which being divided by 6, the index.
of the power gives 5.3010300, which is the Log, of
0.00002, the root of the fixth power required.
And after the fame manner may the root of any other
power be extracted.
5. To find mean proportionals between any two numbers.
Subtract the Log. of the leaft term from the Log. of the
greater, and divide the remainder by a number which
exceeds by one the number of means fought; then add
the quotient to the Log. of the leffer term, or fubtract it
from the Log. of the greateft, continually, and you will
have the Logarithms of all the mean proportionals required.
Example. Let it be required to find three mean pro-
portionals between 106 and 100.
Log, of 106-
·2.0253059
Log. of 100
2.0000000
Divide by
Log. of the first mean
Excefs added
4)0.0253059(0.0063264.75
Log. of the first mean 101.4673846=2.0063264.75
2.0000000
0.0063264.75
Excefs added.
-0.0063264.75
Log. of the fecond
102.95630142.0126529.50
Excefs added
-0.0063264.75
Log. of the third-
104.46704832.0189794.25
Excefs added.
-0.0063264.75
Log, of the fourth-106—
-2.0253059
The use of the tables of Logarithmic fines, tangents and fecants.
1. To find the fine, tangent and fecant of any arch to
ninety degrees.
f
Find
An APPENDIX.
383.
Find the number of degrees at the head or foot of the
table; and the minutes in the left or right-hand column,
and in the common angle of meeting, formed by the title
placed at the head or foot of the column, and the minutes,
you will find the fine, tangent or fecant required.
But if the given arch contains any parts of a minute, in-
termediate to thofe found in the tables; take the difference
between the fines, &c. of the given degrees and minutes,
and of the minute next greater.
Then as I minute is to that difference, fo is the given
intermediate part of a minute in decimals to a fourth pro-
portional, Which being added to the fine, &c. firſt found
will give the fine, &c. required.
Example. The Log. fine of 1°. 48'. 28". 12", is required.
The Log. fine of 1°. 49'. is
The Log. fine of 1°. 48′. is
8.5010798
8.4970784
Their difference is
40014
Then as 1 : 40014:: 0.47 (28″. 12#.): 18807 nearly.
Therefore to the Log. fine of 1°. 48′.
Add the difference found
The fum is the Log. fine of 1°. 48′. 28″. 12″
8.4970784
40014
8.5010798
2. To find the arch anfwering to any given fine, tangent,
or fecant.
Take the next leſs found in the tables, and fubtract it from
that given, obferving the degrees and minutes anſwering
to it, and divide this remainder by the difference between
it and the next greater, adding as many cyphers as are
neceffary to the dividend, and the quotient will be the
decimal part of a minute to be added to the degrees and
minutes before found.
Example 1. Suppofe it were required to find the arch
anfwering to the Log, fine.
9.8393859
The next lefs is the Log. fine of 43°. 41'.-9.8392719
The difference is
1140
Alfo the difference between the Log. fine of 43°. 41′,
and 43°. 42'. is 1322.
Then 1322)1140.000(0.86251.43, which being
added to 43°. 41′, gives 43°. 41'. 51". 43", for the arch
required.
Example
384
An APPENDIX.
ponding to the Log. tangent
Example 2. Let it be required to find the arch corref
The tang. of 25°. 24', the next lefs is
The difference is
9.6766687
9.6765426
1261
Alſo the difference between the Log, tangent of 25°. 24′•
and the Log. tangent of 25°. 25. is 3260.
Then, 3260)1261.0000(0′. 387— 23". 13″".
Therefore 25°. 24. 23". 13" is the arch anfwering to
the given tangent.
F IN I S
}
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