electrogenesis who knew
3141592/24/360=363.61025463
INDEX.
Month dayper hour minute second
Mississippi
1
Min 60
Hr 360
2
δαYpΓ 86400
3
Μοντγ 2592000
4
ΣΟΛ 31415926
mississippi 31
Piθ3
Mississi Mississippi in a Mississippi
Where The tree ball confabulator ater
Taking the mobius strip stripping it ripping it in to the Science of the Word and the Art of the Other side where the science looks like art and art becomes science and τρλιγιον Γελιγιουσλγ goes Softly into the nicht of ΒΓΟΚΕΝ γλώασσ του έχου ανδ έχουμε ανδ έχουσ έχουσ έχους έχουε έχουτ
έχουυ έχουι έχουο έχουα έχουη έχουχ έχουω έχουν έχουμ
Pythagorean ism
https://www.britannica.com/topic/number-symbolism/Pythagoreanism
According to this book of accepted accuracy, the results that
This idaηθ produces are mythical nonsense When numbers are symbols and symbols are numbers the numb
Nuts are going to go bonkers for it...
People also ask
What cause a brain startbbe?
It's during this idle state of mind, “when your brain is shutting down some of its processing systems it believes you don't need in that moment, that brain farts are likely to occur.” Your walk up the stairs or into your neighborhood grocery store, your drive home … these are things your brain recognizes as repetitive ...May 12, 2016
Those are the ones who thing βελιενε there is a difference between their ass and a hole in the ground But are not sure which is which so it is a little confusing ραγτιςυλαγλγ ωηεη η ις ηοτ Η αηδ ω ισ τηε θνΕΝ
That is why is makes no sense to meditate and let your brain organize itself...Better to listen to the smart people telling you to Not believe any difficult concept more complex then the magically scientific comlpexly simple algorithmic solution proposed to the dream tellers told in secret by a mythical fairy with applicability directly to your specific unstated problem anything αςςcepted except to believe in nothing οτηεγ τηαν what you can experience directly repetitively and the others who tell you toο βε βεττεγ βεττγ αηδ Believe in one thing and not believe in no no thing which is just another thing thingly...
Not a lot of choices left in the in f in it e +1 box of ideas that the illusion has painfully painted Ass ωΩΜεφα half full of complex concepts complex numbers such as cents and percents you are smartly told by the nearest shiny penny grabbing cheek pincher imaginary numbers representing imaginary complex concepts such as rubber balls rubber ducks everything on the one side created by the line defined by the now arbitrary other side well defined within the background set space defined by the definition of said space of sets
and half full of non sense whence the explanation careens career ward carefully cosining tangentially the free sine sitting idly by perpendicularly tan that you π
Προυνουνςεδ πεε σεε Σιγματιος
Shake stir and watch the smoke rise
ΑΣ ιτ ιηενιταβλγ ωιλλ ξγθμ ενεγγ
Μολτεη ΣΔαλτ Γεαςυθγ ΣνεΓυωηεγε
Αλλ Ωαγσ where 在哪里 ΟΠΟΥ τηε Δαθ ωηιςη ις ηαμεδ ις δοζ and a dog by any other name is still a dog
Bestimate from the happy farm if a day be considered 86400 mississippi
It can also be 386400 babbiieeiiss
386400
166400 -
120000 =
Net net the babbiieeii blue river Styx
Adds two minutes of mississippii
Every nano day of the big messyyssyyppyy
Some days referred to as a day
Mountain inglese to
365 X 120 = 43800 Mississippii
minηθωσs added annually to the flow with the go
Electrogenetically
Electrogenesis are the closely-related biological abilities to perceive electrical stimuli and to generate electric fields. Both are used to locate FOOD...strongE electric discharges are used in ωατερ if you can believe it in groups of fishe to stun OTHER FOOD not everyone like the emp blew out the dock and every machine in side except the manual bullet babies. The capabilities are found almost exclusively in aquatic or amphibious animals, SUCH AS FISH since water is a much better conductor of electricity than air. HAHAHA Who thought Air was water when it was snowing or not rain manning...As bozo often famously remarked...some people did not know this In passive electrolocation, food objects are detected by sensing the electric fields created by said food as said food digests other food and the electric fish perfer sushi au naturel... In active electrolocation, fish use electrosensory sense mode to generate a weak electric field stronger than the food field but weaker than the food detector field detector and sense the different distortions of that field interacting through the matrix of solis air not NoH2 no no clean lean bluegreen H2o as objects that conduct and resist electricity conduct and resist eelectricity in the immediate vicinity of said hungry for sushy fishy Active electrolocation is practised by two groups of actively outwardly and inwardly everready electric fish she Gymnotiformes electriccarvingknifefishe and he Mormyridae electricelephantfishe and by Gymnarchus niloticus, the African knifefish...All electric fish generate an electric field using an electric organ, located in muscles in the end that tails the discharge of stunny juice...The field is called w
if it is enough to detect prey and AW if it is powerful enough to stun or kill the dish...The field may be in brief pulses, as in the elephantfishes, or a continuous wave, as in the knifefishes. Electric fish, such as the electric eel, locate prey by generating constant electric fields dissipating those fields as sonar testing the waters as the shiny pennies loathe to slay and then when the menu has been perused pointillistically pow eddy te eel slaps zaps and poof a discharge of electric current sharpe ar tha the arrow of accurately Achilles the archer strongly enough to stun the prey not themselves or the entire content of the warn stlitz bath they are being monitored in to collect this electrifying informative mosaic of mundane Monday morn8ng mealtim3 other strongly electric fish such as the electric ray electrolocate passively.
The stargazers are unique in being strongly electric but not using electrolocation...See stargazers look through the empty atmosphere of O2NCEtc around the invisible sneaky rona bits and color the sky candy clown crayon crabbed rays of redshirt matriziny past go directly to whorls of wheeled color angularly momentalized mere minutes away from others also known as oxidizing agents in the infinite ground of loosely organized synchronicity that the world out there be working one way from atom to zee
Angles, plane vertically opposite,
46.
regular solid, 630 solid vertically opposite , 503.
solid , measurable by spherical area, 501.
Acute-angled triangle, 12 Algebra and Geometry connected,
134.
Aliquot part, definition of, 327.
Alternate angles, definition of, 75 .
arc, definition of, 244.
Altitude, 351.
Ambiguous case, preface, x.
solid, mode of denoting, 501.
Anharmonic pencils, having com mon ray , 439.
transversals of, 437.
property of four points on circle, 440 property of four planes, 640.
property of four tangents to circle, 441.
ranges, having common point,
438.
ratio of four points on circle,
442.
property of, 436.
Antecedent, definition of, 328. Apollonius, circle of, 426 Arc, definition of, 167, Archimedes, 644.
Arcs, direction of measurement of, 289.
Area, definition of, 10.
Arithmetical Progression , 462.
Away from , why used , 33, 41.
Axiom , use of term , preface, vi .
Axioms, definition of, 15.
examples of, 15. the list of, imperfect, preface,
vii.
Axis of circle, 507 of similitude defined , 454
Axis of triangles, 444.
radical, 264 of pair of circles touching three circles, 456.
Axes, spheres of similitude of three, 595.
- of similitude of three circles,
454.
Baseof parallelogram , 351.
of triangle, 351.
Brianchon's theorem, 447.
Casey's theorem , 469-471.
Centre, of circle, 13. Cf. 174.
of direct similitude, when infinitely distant, 454.
of picture, 577, 578.
radical, 265 . - of similitude, (a) direct, ( 6) inverse, 455.
of sphere, 507.
Centres of similitude of four spheres, relative positions of,
596.
Centroid, centre of gravity, de finition of, 103.
of weights, 425, 584. Ceva, theorem of, 422, 584.
converse of, 424.
Chord, definition of, 167.
of arc, 167.
circle, 167.
contact of circle touching two circles, 451.
Chords of two circles intersecting
on radical axis, 452.
Circle, definition of, 13.
escribed, definition of, 294.
inscribed, definition of, 172,
294.
of Apollonius, 426.
Nine-point, 271.
great, of a sphere, 590.
small, 591.
Circles, coaxial, definition of,
429.
- construction forcircles to touch
three, 458.
- equal, have equal radii, 231.
Circles, four pairs of circles touch three, 459.
orthogonal, 266.
pair of, touching two circles,
455.
touching three circles, 455.
touching three circles, radical axis of pair of, 456 .
Circumference, definition of, 167.
Circumscribed circle, definition of,
297. Cf. 172.
Coaxial circles, properties of, 268,
465, 479, 481, 482, 483.
triangles, definition of, 444.
compolar, 445.
Collinear points, 351.
Commensurable, definition of,
328.
Common section of two planes,
definition of, 500.
Complanar spheres, definition of,
594.
Complete quadrilateral, 353.
Complement. Complementary angles, definitions of, 45.
Complements of parallelograms,
&c . , 111 . Compolar triangles, definition of, 444.
coaxial, 444.
Concentric, definition of, 170.
Concurrent lines, definition of,
351.
examples of, 53, 71 , 95,
103, 422, 424, 582, 583,584,587.
Cone, circular , 507.
right circular, 507.
vertex of, 507.
enveloping, of sphere, 592.
Configuration of three circles touching a fourth circle, 459.
of four circles touching a fifth circle, 471.
Conjugate, 648.
points, 352 .
-
>
rays, 352. Consequent, definition of, 328.
Contact, external, 199.
internal, 201.
Contain . Arcs contain angles, 168.
Contained, a rectangle is , 134.
Continuity, principle of, 472,
473, 474.
Converse propositions, 27.
not necessarily true, 27.
Convex figure, plane, definition of, 10.
solid, definition of, 505.
Corner, 505.
Correspondence of sides of tri angles, 256.
Corresponding angles, definition of, 75 .
vertices, 349.
Cube, definition of, 505, 506.
construction of, 634.
Curve, inverse of, 460.
Curves, angle between, 266.
Cylinder, circular, 507.
right circular, 507.
-- enveloping, of sphere, 592.
Each to each , why discarded, pre face, ix.
Enunciation , joint for Props. 5 and 6, Book II. , 145.
- joint for Props. 9 and 10, Book II., 153.
Equal in all respects, definition of, 13.
plane figures, 13.
solid angles, 502.
solid figures, 506.
in opening, 502.
Equality of ratios, definition of,
330.
remarks on, 329.
ratio of, 328.
Equiangular triangles, property of sides of pair of, 256.
- polygons, 506.
Equilateral figure, 11 .
Equimultiples, definition of, 327.
Euclid, by his Postulates restrict
ed himself in his use of instru .
ments, 21.
in Prop. 4, Book I. , assumes that two lines cannot have a
common part, 23.
Euler's equation, 629.
Example of the principle of con tinuity, 474.
Exterior angles, definition of,
75.
Extrascribed sphere of tetra hedron, 628.
Extreme and mean ratio, 351.
Extremes, definition of, 332.
Data, definition of, 61 .
often conditioned, 61 .
Decagon, definition of, 285.
Desargues, theorems due to, 444,
445.
Diagonal, of parallelepiped, 562.
Diagonals, of a complete quadri lateral, 353.
definition of, 10.
Diameter, of circle, 13.
of sphere, 507.
bisects circle, 175.
Direction of measurement of arcs,
289.
Distance from a point, definition of, 171.
between centres of inscribed
and circumscribed circles of
triangle, 476.
Distances of point from two points, sum of multiples of squares on , 478.
Dodecagon, definition of, 285.
Dodecahedron, 505.
regular, construction of,
636.
Duplicate ratio, definition of,
332,
Figure, definition of, preface, vi.
circumscribed, 172, 285.
convex, definition of, 10.
equiangular, 11.
equilateral, 11 .
inscribed, 172, 285.
plane, 6 .
rectilinear, 9 .
regular, 11.
skew , 504.
Figures, impossible, preface, ix.
reflex, 621.
reverse, 561.
Figures, reversion of, caused by reflection , 620, 621, 623.
symmetrical, 623.
translation of, 615,
Gauss, what regular polygons can be inscribed in a circle, 320.
Geometrical Progression , 462.
Geometry, plane and spherical compared, 606.
solid , definition of, 499.
Gergonne's construction for cir cles touching three circles,
458 .
Gnomon, why discarded , preface,
vi.
Harmonic pencils, having com mon ray, 435.
transversals of, 433.
property of polar, 430.
range, defined , 352.
property of, 428, 432. ranges, having common point,
434.
Harmonical Progression , 462. Hexagon , definition of, 285.
Hexahedron , 505.
regular, construction of, 634.
Homologous, definition of, 331.
Hypotenuse, definition of, 11.
Inverse curves, definition of, 460.
mechanical mode of draw ,
ing, 468.
ofcircle through pole, 463,
of circle not through pole, 463,
of straight line, 462.
surfaces, 600 .
of plane through pole, 600.
not through pole, 600.
of sphere through pole, 600.
not through pole , 600.
Inversion, definition of, 460, 600 .
angle between two curves not altered by, 461, 601.
distance between two points in terms of inverse points,
464.
- coaxial circles invert into co
axial circles, 465.
concentric circles invert into
coaxial circles, 465.
concentric spheres invert into complanar spheres, 601.
- complanar spheres invert into
complanar spheres, 601.
limiting points invert into limiting points, 465, 601 .
- locus of point P, where mPA = nPB, 464.
one circle may be its own in
verse, 466.
- pole of, 460, 600.
Ptolemy's theorem proved by,
467.
radius of, definition of, 460,
600.
two circles may be their own inverses, 466.
two circles, how inverted into equal circles, 466.
two points and their inverses lie ona circle, 461.
two spheres may be their own inverses, 601 .
three spheres may be their own inverses, 601 .
two spheres may be inverted with two equal spheres, 601.
three circles may be their own inverses, 466.
Inversion , three circles, how in verted into equal circles, 466.
three spheres may be inverted into three equal spheres, 601 four spheres can be inverted into four equal spheres, 601.
Isosceles, definition of, 11.
Maximum value, example of, 190 ,
192, 193 , 195.
Maximum and minimum, illus tration of, 55 .
and minimum values occur alternately, 193.
Mean proportional, definition of,
332 .
Means, definition of, 332.
Measure, definition of, 327.
Menelaus, theorem of, 418, 584.
converse of, 420.
extension of, 641.
Method of superposition, 5.
Minimum value,definition of, 55.
-example of, 55, 57, 193, 195.
Multiple, definition of, 327.
- definition of mth,3 327.
Letters, used to represent magni.
tudes, 327.
Like anharmonic pencils, 353.
anharmonic ranges, 352.
Limit, explained, 217.
Limiting points of a series of co.
axial circles, 429.
Line, definition of, 2.
horizon, 577.
of sight, 577. pedal, definition of, 272.
Simson's, 272.
straight, definition of, 4.
tangent to sphere, 592.
Lines, additionof, 6 .
cut proportionally (1) extern .
ally, (2) internally, 350.
- dotted, use of, 499.
- polar, with respect to sphere,
593.
Locus, definition of, and example,
39.
ofpoints, distances from which to two fixed points are equal,
39.
- distances from which to two fixed points are in a fixed ratio ,
426, 464.
tangents from which to two circles are equal, 264.
tangents from which to two circles are in a fixed ratio,
479, 480, 481 .
Lune, definition of, 602 .
angle of, 602.
area of, 603.
Nine- point circle, definition of,
271 .
properties of, 270, 271,
448, 449.
Notation of proportion, 330.
Obtuse -angled triangle, definition of, 12.
Octagon, definition of, 285. Octahedron, 505.
- regular, construction of, 635.
Orthocentre, definition of, 95.
of tetrahedron, 588 Orthocentric tetrahedron, 588 .
- twelve -point sphereof, 588.
Orthogonal circles, 266-268.
Orthohedron, 506.
Orthogonally, system of circles cutting two circles, 268.
a Parallel, definition of, 7.
lines, 7.
line, to plane, 504.
- plane, to plane, 504.
Parallelepiped, definition of, 506.
diagonal of, 562.
general properties of, 582.
- rectangular, 506.
special forms of, 586.
aīlied tetrahedrons of, 582,
586.
Parallelogram , definition of, 12.
Parallelograms about the dia
gonal of a parallelogram , 111.
sections of tetrahedron, 585.
Part, definition of mth, 327 .
Parts of a triangle, definition of,
74.
Pascal's theorem , 446.
Peaucellier's cell, 468.
Pedal line, 272.
Pencil, anharmonic, 352.
definition of, 351.
- how denoted, 351 .
- harmonic, 352.
– like anharmonic, 353.
Pencils, equality of, test of, 440.
Pentagon, definition of, 285.
regular, construction for, 309 .
Pentahedron , 505 Perimeter, definition of, 9.
Perpendicular, definition of, 9.
Perspective, triangles in , 444.
- principles of, 576—581.
Picture, centre of, 577.
plane , 576.
Plane, definition of, 6, 500 .
— polar, of point with respect to sphere, 593.
tangent, to sphere, 592.
- radical, of two spheres, 593.
Planes, common section of two,
500.
test of the coincidence of two,
500.
– inclined to one another, 504.
- parallel to one another, 504.
of similitude of four spheres,
596.
Point, angular, 9, 505.
definition of, 2.
pole of plane with respect to sphere, 593.
vanishing, 578.
Points, conjugate, defined , 352.
limiting, of a series of coaxial circles, 429.
Polar, definition of, 258. lines with respect to sphere,
593.
Pole, definition of, 258.
Pole, of inversion, 460.
of line with respect to circle,
258.
of plane with respect to sphere,
593.
of triangles, 444 .
Polygon , definition of, 11 .
the term extended to include triangles and quadrilaterals,
349.
Polygons, equiangular, 349.
number of conditions of equi.
angularity of, 349.
—number ofnecessary conditions of similarity of, 350.
- regular, which can be inscribed in a circle, 320.
– similar, 349.
Polyhedron, 505.
convex, definition of, 505 .
regular, 505.
closed, relation between faces,
edges and corners, 629.
Polyhedrons, equiangular, 506 .
similar, 506 .
Poncelet's theorems, 483.
Porism , definition of, 475. - examples of, 475–477.
of inscribed and circumscribed
circles, 477.
condition for, 476.
- of set of coaxial circles, 483.
Postulate, definition of, preface,
viii, 4.
- I. Two straight lines cannot enclose space, 4. - II. Two straight lines cannot have a common part, 4.
- III. A straight line may be drawn from any point to any other point, 5.
- IV. A finite straight line may
be produced to any length , 5.
- V. All right angles equal,
preface, viii, 9.
-VI. A circle may be described with any centre, and with any radius, preface, ix, 14 .
- VII. Any straight line drawn
figure must, if produced far enough, intersect the figure in two points at least, 14.
Postulate VIII. Any line joining two points one within and the other without a closed figure must intersect the figure in one
point at least, 14,
- IX . If the sum of the two in teriorangles, which two straight lines make with a given straight line on the same side of it ,be not equal to two right angles,
the two straight lines are not parallel, 51.
Book XI.
I. A plane may be drawn through any three points, 500.
II. A part of a plane may be produced to any extent in any direction in its plane, 500.
Principle of continuity, definition of, 472.
examples of, 474.
Prism , 506.
Problem, definition of, preface, xi.
Problems often admit of several
solutions, 17 , 19, 249, 287.
Progression, Arithmetical, Geo metrical, Harmonical, 462.
Projection, definition of, 533 .
Proportion, proportionals, defini tion of, 330.
- continued, definition of, 332.
notation of, 330.
Proportional, reciprocally, defini tion of, 380.
Proportionally cut, externally,
350 .
- internally, 350.
Ptolemy's Theorem, property of chords joining four points on circle , 257 .
Casey's extension of, 469,
470, 471 .
employed, 636 , 638.
Pyramid , 506 .
Pythagoras, Theorem of, 120.
Quadrilateral, complete, 353.
Radical axis, of two circles, 264.
of three spheres, 594 .
centre, of three circles, 265.
of four spheres, 594.
plane of two spheres, 594.
Radius, of circle, 13.
of sphere, 507.
of inversion, 460.
vector, definition of, 460.
Range, anharmonic, 352.
definition of, 351.
how denoted, 351.
Ranges, like anharmonic, 352 .
Ratio of anharmonic range, 352,
443.
compounded, definition of,
332.
independent of order of composition, 399.
- definition of, 327.
- of equality, 328.
- of greater inequality, 328.
of less inequality, 328.
of a pencil, 353.
of ratios, 332.
Ratios compounded, 332.
equality of, defined, 330.
Rays, conjugate, 352.
of a pencil defined , 351.
Reciprocal figures, 607.
Reciprocally proportional, 379.
Rectangle, definition of, 12 , 134.
how denominated , 134.
Re- entrant quadrilaterals, 353.
Reflection, in point, 620.
in straight line, 621.
in plane, 623.
Reflections, successive, in planes,
623.
in points, 620.
in straight lines, 622.
Reflexes, definition of, 620, 621 ,
623 .
Regular plane figure, definition of,
11 .
solid angles, 630.
- solids, only five possible, 631 .
solids, construction of, 632— 639.
- Dodecahedron, 636.
Hexahedron (Cube), 634.
Icosahedron , 638 .
Octahedron , 635.
Tetrahedron , 633.
Relations between a line and a
circle, 213.
– between two circles, 203.
Respectively, how used,preface, ix .
Reverse tetrahedrons, 561.
trihedral angles, 561.
Rhomboid, why discarded, pre face, vi.
Rhombus, definition of, 12. Right angles are equal, 37. Cf. preface, viii.
at, line and line, 9.
line and plane, 504.
plane and plane, 503. Right- angled triangle, definition
of, 11.
Rotation in plane point about point, 186 , 614. line about point, 186,
614.
- plane figure about point,
186, 589, 615.
in space, point about line, 616.
line about line, 616.
figure about line, 589,
616.
Rotations, successive, in a plane,
618.
in space, 619.
solid figures, 595 .
axes of, for three spheres, 595.
- planes of, for four spheres, 596.
centres of, of two circles, 450.
of three circles, 454.
of circle of nine points and circumscribed circle, 449.
Simson's line, 272.
Skew , figure, definition of, 504,
- quadrilateral, sphere touching four sides of, 624—627.
Solid, definition of, 3. geometry, definition of, 499.
Solids, regular, 632–639.
definition of, 505.
names of, 505 .
Sphere, surface of, 642, 643.
- volume of, 644. - touching four spheres, 598
599,
description of, 598.
inscribed in tetrahedron , 608—
613.
definition of, 507.
properties of, 590—599.
how denoted, 507.
Twelve -point, 588.
great circle of, 590 .
- tangent line to , 592.
plane to, 59
Reflexes, definition of, 620, 621 ,
623 .
Regular plane figure, definition of,
11 .
solid angles, 630.
- solids, only five possible, 631 .
solids, construction of, 632— 639.
- Dodecahedron, 636.
Hexahedron (Cube), 634.
Icosahedron , 638 .
Octahedron , 635.
Tetrahedron , 633.
Relations between a line and a
circle, 213.
– between two circles, 203.
Respectively, how used,preface, ix .
Reverse tetrahedrons, 561.
trihedral angles, 561.
Rhomboid, why discarded, pre face, vi.
Rhombus, definition of, 12. Right angles are equal, 37. Cf. preface, viii.
at, line and line, 9.
line and plane, 504.
plane and plane, 503. Right- angled triangle, definition
of, 11.
Rotation in plane point about point, 186 , 614. line about point, 186,
614.
- plane figure about point,
186, 589, 615.
in space, point about line, 616.
line about line, 616.
figure about line, 589,
616.
Rotations, successive, in a plane,
618.
in space, 619.
solid figures, 595 .
axes of, for three spheres, 595.
- planes of, for four spheres, 596.
centres of, of two circles, 450.
of three circles, 454.
of circle of nine points and circumscribed circle, 449.
Simson's line, 272.
Skew , figure, definition of, 504,
- quadrilateral, sphere touching four sides of, 624—627.
Solid, definition of, 3. geometry, definition of, 499.
Solids, regular, 632–639.
definition of, 505.
names of, 505 .
Sphere, surface of, 642, 643.
- volume of, 644. - touching four spheres, 598
599,
description of, 598.
inscribed in tetrahedron , 608—
613.
definition of, 507.
properties of, 590—599.
how denoted, 507.
Twelve -point, 588.
great circle of, 590 .
- tangent line to , 592.
plane to, 59
623 .
Regular plane figure, definition of,
11 .
solid angles, 630.
- solids, only five possible, 631 .
solids, construction of, 632— 639.
- Dodecahedron, 636.
Hexahedron (Cube), 634.
Icosahedron , 638 .
Octahedron , 635.
Tetrahedron , 633.
Relations between a line and a
circle, 213.
– between two circles, 203.
Respectively, how used,preface, ix .
Reverse tetrahedrons, 561.
trihedral angles, 561.
Rhomboid, why discarded, pre face, vi.
Rhombus, definition of, 12. Right angles are equal, 37. Cf. preface, viii.
at, line and line, 9.
line and plane, 504.
plane and plane, 503. Right- angled triangle, definition
of, 11.
Rotation in plane point about point, 186 , 614. line about point, 186,
614.
- plane figure about point,
186, 589, 615.
in space, point about line, 616.
line about line, 616.
figure about line, 589,
616.
Rotations, successive, in a plane,
618.
in space, 619.
solid figures, 595 .
axes of, for three spheres, 595.
- planes of, for four spheres, 596.
centres of, of two circles, 450.
of three circles, 454.
of circle of nine points and circumscribed circle, 449.
Simson's line, 272.
Skew , figure, definition of, 504,
- quadrilateral, sphere touching four sides of, 624—627.
Solid, definition of, 3. geometry, definition of, 499.
Solids, regular, 632–639.
definition of, 505.
names of, 505 .
Sphere, surface of, 642, 643.
- volume of, 644. - touching four spheres, 598
599,
description of, 598.
inscribed in tetrahedron , 608—
613.
definition of, 507.
properties of, 590—599.
how denoted, 507.
Twelve -point, 588.
great circle of, 590 .
- tangent line to , 592.
plane to, 59
Sphere, enveloping cone of, 592.
cylinder of, 592.
– touching sides of skew quadri.
lateral, 624.
Spheres, intersection of, 592.
Spherical trigonometry, 602, 606.
triangle, 602.
area of, 603.
sides of, 602 .
- triangles, reverse, 603.
polar, 604.
Square, definition of, 12.
- ordinary definition, why dis carded, preface, vi.
Straight lines, test of equality of, 5.
Superposition, method of, 5 .
note on, preface, viii.
Supplement, supplementary an gles, definitions of, 45.
Surface, definition of, 3.
closed, 504.
- inverse of, 600.
Symmetrical, definition of, 623.
Tetrahedrons, reverse, 561.
Theorem of Apollonius, 426.
Brianchon's, 447.
Casey's, 469, 470, 471.
Ceva's, 422, 584.
of Desargues, 444 , 445.
of Menelaus, 418 , 584.
extension of, 641.
Pascal's, 446.
Poncelet's, 482, 483.
Ptolemy's, 257.
of Pythagoras, 120. Third Proportional, definition of 332.
Touch, meaning of, 169.
Translation , definition of, 615.
Transversal, definition of, 419.
Trapezium , why discarded , pre face, vi .
Triangles, coaxial, 444.
- compolar, 444 .
- definition of, 11.
equal , on, 74.
inscribed in a circle, 416.
missing case of equality of,
preface, x.
Trihedral angle, definition of,
501 .
construction of, 560.
- angles, opposite vertical, 572.
Triplicate ratio, definition of, 332.
Table respecting planes of simili tude offour spheres, 596.
inscribed and escribed
spheres of tetrahedron , 613.
Tables the regular solids, 632,
639.
Tangent, definition of, 169.
common to two circles, 219.
limit of, a secant, 217, 245, 253.
examples of, 245, 253.
Test of equality of plane angles, 8.
of solid angles, 502.
geometrical figures, 5.
straight lines, 5.
Tetrahedron, 505 .
description of a sphere about,
573.
inscribed sphere of, 608 .
general properties of, 583.
special forms of, 586.
orthocentric, 588.
– regular, construction of, 633.
two distinct types of, 612.
intrascribed sphere of, 628.
Tetrahedrons, allied, of a parallel epiped, 582, 586.
Units of length and of area , 135 .
Vanishing point, 578.
- points of three directions mu tually at right angles, 578.
Vertex , 7.
definition of, 9.
of a pencil defined , 351 .
of polygon, 7. polyhedron, 505 .
cone, 507. Volume, of sphere, 644.
tetrahedrons, 568, 570.
Weights, centroid of, 425, 584.
Within a circle, 169.
Without a circle, 169.
In the two right-angled triangles OCP, OCQ,
OP, OC are equal to OQ, OC ;
therefore CP is equal to CQ.
Similarly every other line from C to the section PQR is equal
to CP .
Therefore the section PQR is a circle of which C is the centre.
Definition . The section of a sphere made by a plane not passing
through the centre is called a small circle of the sphere.
4. Equal circular sections are at the same distance from the
centre of the sphere, and circular sections at the same distance from
the centre are equal. (Cf. III. Prop. 14, parts 1 and 2.)
5. Of two plane sections of a sphere the one which is the nearer to the centre is the greater. (Cf. III. Prop. 15 , parts 1 and 2.)
6. Two spheres, which meet one another, must either intersect in a circle whose axis passes through their centres, or touch in a point on the line joining their centres. (Cf. III. Props. 9–13
602 BOOK XI.
ELEMENTS OF SPHERICAL TRIGONOMETRY .
Definition . Each of the four parts into which the surface of a
sphere is divided by two great circles is called a lune.
The angle at which the bounding semicircles intersect is called
the angle of the lune.
It can be shewn by the method of superposition that on equal spheres lunes of equal angles are equal; and hence that the areas of
two lunes of different angles are in the same ratio as the angles of
the lunes. Hence the area of a lune of angle A is to the area of the sphere in the same ratio as A to four right angles.
A
B
B'
A '
Let OABC be a trihedral angle, and let ABC lie on a sphere whose centre is 0. The triangle ABC which is formed by the three arcs
BC, CA , AB of great circles of the sphere, is called a spherical
triangle : and as the tangents to the arcs AB, AC at A are at right angles to OA, the angle between them, which is called the angle A of the triangle BAC, is the inclination of the faces OBA, OCA.
The angles BOC, COA, AOB, which are in the same ratio as the
arcs BC, CA, AB of a great circle, are called the sides of the
spherical triangle ABC and are denominated a, b, c respectively.
We can therefore at once translate the theorems which were
established with reference to the plane angles and the inclinations
of the faces of a trihedral angle into theorems with reference to the sides and the angles of a spherical triang
SPHERICAL TRIGONOMETRY. 603
a
>
Thus we may state that in a spherical triangle ABC whose angles are A, B, C and whose sides are a, b, c,
( 1 ) If a is equal to b, A is equal to B. (Ex. 3, p. 527.)
(2) If A is equal to B, a is equal to b.
( 3) If a is greater than b, A is greater than B. (Ex. 4, p. 527. )
(4) If A is greater than B, a is greater than b.
( 5) The sum of any two of the sides a, b, c is greater than the third side. (Prop. 20.)
( 6) The sum of the sides a, b, c is less than four right angles.
( Prop. 21 , Part 1. )
If a sphere whose centre is O cut the faces of a trihedral angle QABC in the spherical triangle ABC, and also cut the faces produced beyond 0 in the spherical triangle A'B'C ' , the spherical triangles ABC, A'B'C ' have their corresponding sides and their corresponding angles equal in pairs but it is not possible to shift either of the triangles so as to coincide with the other : two spherical triangles
whose sides nd angles are equal in pairs but which cannot be made
to coincide are called reverse.
Since the area of a lune of angle A is to the area of the sphere in
the same ratio as A to four right angles ; therefore if the ratio of a
to 1 be equal to the ratio of A to four right angles the area of the
lune of angle A : the surface of the sphere = a : 1, or the area of the
lune of angle A = as, where S represents the area of the surface of the sphere. Now take a spherical triangle ABC and let the ratios
of the angles A, B, C to four right angles be a to 1, B to 1 , y to 1
respectively ; let the great circles of which the sides of the triangle
form parts be completed, they will divide the surface of the sphere
into four pairs of reverse triangles. Let the reverse triangle of ABC
be A'B'C' .
Then the lune ABAC = aS ;
the lune BCB'A = BS ;
the lune CAC'B = yS ;
and the area of the triangle BAC' is equal to the area of the reverse triangle CA'B' , ( Add. Prop. 17. )
therefore the sum of these three lunes is equal to half the surface of
the sphere together with twice the area of the triangle ABC.
Hence, IS + twice the area of the triangle ABC= (a + B + y) S. Or the area of the triangle ABC = } (a + B + 7-3) S..
604 BOOK XI.
Let ABC be a spherical triangle; let the axis of the great circle
BC cut the sphere in P and P' , in Spherical Trigonometry P and P '
A
P
R'
R
are called the poles of BC. Let P be the pole which lies on the same side of BC as A , which is equivalent to saying the arc PA is
less than a quadrant ; similarly let Q and R be the poles of the great circles CA , AB which lie on the same side of them as B, C, respec
tively. Then the spherical triangle whose vertices are P, Q, R is
called the POLAR TRIANGLE of ABC.
OQ and OR are at right angles to the planes OCA, OAB, therefore
the angle QOR is the supplement of A the inclination of the planes
OCA, OAB. Similarly the angles ROP, POQ are supplements of the angles B and C.
Now because Q is the pole of CA , the arc AQ is a quadrant, and
because R is the pole of AB, the arc AR is a quadrant, and because
the arcs AQ and AR are quadrants, A is a pole of QR. A is on the
same side of QR as P, because the arc AP is less than a quadrant.
Similarly B, C are poles of RP and PQ and on the same side of them
as Q and R respectively.
SPHERICAL TRIGONOMETRY. 605
Hence if PQR be the polar triangle of ABC, then ABC is the polar
triangle of PQR, and the sides of each are the supplements of the angles of the other.
In the polar triangle PQR any two sides are together greater than
the third, i.e. q + r is greater than p, therefore in the original triangle
ABC, the sum of the supplements of B and C is greater than the
supplement of A; or, A + two right angles is greater than B + C ;
therefore in a spherical triangle any one of the three angles together with two right angles is greater than the sum of the other two angles.
Now in the polar triangle PQR, the sum of the sides p, q, r is less than four right angles (Prop. 21, Part 1) ; therefore in the triangle ABC, the sum of the supplements of the angles A , B, с
is less than four right angles ; or A + B + C, that is the sum of the
three angles of a spherical triangle, is greater than two right angles.
Two triangles ABC, A'B'C ' on the surface of the same sphere are
either equal in all respects or reverse
(I) If a =a' , b = b' and C= C' . (Add . Prop. 2. )
(II) If a = a' , b = b' and c = c'. (Add. Prop. 1.)
(III) If a = a', b = h' and A = A ' and if in addition
( 1) B and B' be each less than a right angle,
or ( 2) B and B' be each greater than a right angle,
or (3) either B or B' be a right angle. (Add. Prop. 3. )
And hence in consequence of the properties of the polar triangle,
(IV ) If A =A',> B=B' and c =c' . ( Add. Prop. 2.)
(V) If A = A', B =B' and C= C' . ( Add. Prop. 1. )
(VI) If A = A ', B= B' and a = a ' and if in addition
( 1) band ' be each greater than a right angle,
or (2) b and b' be each less than a right angle,
or ( 3) either b or b' be a right angle. ( Add. Prop. 3. )
606 BOOK XI.
It will be observed that the geometry of great circles on the surface
of a sphere differs in several important particulars from the geometry
of straight lines in a plane.
Whereas, in a plane triangle, the sum of the angles is always equal
to two right angles, in a spherical triangle, the sum of the angles is always greater than two right angles.
Whereas in a plane triangle, when two angles are given, the third
angle and the shape but not the size of the triangle is determinate ;
in a spherical triangle when two angles are given the third angle and the shape and the size of the triangle are all indeterminate, but
three angles determine both the shape and the size of the triangle .
Again, whereas in a plane triangle all equilateral triangles are similar and have angles of the same size, the angles in an equilateral spherical triangle may have any value between two thirds of a right
angle and two right angles, the smaller value occurring when the
triangle is extremely small, and the larger value, when the triangle approximates to a great circle. 
The vast majority (99.86%) of the system's mass is in the Sun, with most of the remaining mass contained in the planet Jupiter. The four inner system planets—Mercury, Venus, Earth and Mars—are terrestrial planets, being composed primarily of rock and metal. The four giant planets of the outer system are substantially larger and more massive than the terrestrials. The two largest, Jupiter and Saturn, are gas giants, being composed mainly of hydrogen and helium; the next two, Uranus and Neptune, are ice giants, being composed mostly of volatile substances with relatively high melting points compared with hydrogen and helium, such as water, ammonia, and methane. All eight planets have nearly circular orbits that lie near the plane of Earth's orbit, called the ecliptic.
Earth orbits the Sun at an average distance of 92,960,000 miles
in a counterclockwise direction as viewed from above the Northern Hemisphere.
Given the Euclid explanation of radii triangles and bent straight lines 9296*2*3.141592=584,084,784.64
miles representing the cicumfrence of the big circle
One complete orbit takes 363.61025463 days (1 real year), during which time Earthe travele ~584,000,000 milese eh?
as it turns out it takesa 3,141,592 seconds for the eartha to makae 363.6 incomplete rotations around itself and one complete rotation around the sunny bunny
31415925/(24*60*60)=363.61025463 for referemce
3.1415926535897932384626433832795/(24*60*60)=0.00003636102
you float your boat as the tide calls anyway anyone cuts it the operator operates without anesthetic
In this description of Arithmetic and Algebra set theory as embedded in the operation theories is readily apparent. Where mathematics is set theory this is this:
In Arithmetic or in Algebra, to represent a given length,
take a definite length , for instance an inch
as a unit of length
and express the given length
by the number of units of length
contained in the superset called length
In the same way
if we wish to represent a given area
take a definite area
for instance a square inch
as a unit of area
express the given area by the number
of units of area
contained in the superset of areas called area.
If a rectangle have 2 inches in one side
and 3 inches in an adjacent side
, its area consists of 2 x 3 or 6 squares
each square having one inch as its side
similarly
if a
rectangle have m units
of length in one side
and n units
of length in an adjacent side
its area consists of mn squares
each square having
a unit of length as its side.
Thus in Arithmetic or in Algebra
the area of a rectangle
is represented by the product of the numbers
, which represent
the lengths of two adjacent sides.
If the rectangle be a square
its area is represented by the square
of the number of units
which represents the length of a side
In consequence of this connection
between Algebra and Geometry
there is a certain
corresponding correspondence
between the theorems and the problems
of the Second Book of Euclid
and the theorems and the problems
in Algebra
A short statement
of a corresponding proposition in Algebra
is given as a note to each Proposition
in which statement
each straight
line is represented
by a corresponding letter m or n
and each area
by a corresponding product mn