- Bonaventura Francesco Cavalieri was an Italian mathematician born in 1598 who developed the method of indivisibles, a precursor to integral calculus. - His method treated areas as composed of lines and volumes as composed of planar areas, allowing him to solve problems involving areas, volumes, and centers of mass. - Though crude, his method provided practical techniques for computing areas and volumes and influenced later mathematicians like Kepler and Galileo.

1. Bonaventura ,Cavalieri
“Rigor is the concern of philosophy not of geometry.”
– Bonaventura
Cavalieri

2. Bonaventura Francesco Cavalieri
 Italian priest and mathematician
 Born at Milan in 1598,
 died at Bologna on November 27, 1647.
 He became a Jesuit at an early age;
 on the recommendation of the Order he was in 1629 made professor of
mathematics at Bologna; and he continued to occupy the chair there until
his death. I have already mentioned Cavalieri's name in connection with the
introduction of the use of logarithms into Italy, and have alluded to his
discovery of the expression for the area of a spherical triangle in terms of
the spherical excess.
 He was one of the most influential mathematicians of his time, but his
subsequent reputation rests mainly on his invention of the principle of
indivisibles.

3.  Born Francesco Cavalieri in Milan, he took the name of his
father, Bonaventura, when he entered the Congregation of
Hieronymites or Jesuati (not Jesuit) order when he was
thirteen. The order was established in 1367 to care for and
bury the victims of the Black Death, the plague that killed
more than one-fourth of Europe’s population.
 He received minor orders in 1615, and the next year he was
transferred to the Jesuiti monastery at Pisa. There he studied
philosophy and theology and was introduced to geometry, to
which he devoted the rest of his life. He quickly absorbed the
works of Euclid, Archimedes, Apollonius and Pappas.
Cavalieri became an accomplished mathematician and one of
the most illustrious disciples of Galileo.

4.  He was one of the most influential mathematicians of his
time, but his subsequent reputation rests mainly on his
invention of the principle of indivisibles.
 Indivisibles are difficult to explain precisely. We can
think of them as in some sense the things from which
continuous substances are constructed. A point was the
“indivisible” of a line, that is, Cavalieri considered a line
to be composed of an infinite number of points. A plane
is composed of an infinite number of indivisibles, namely
lines, and squares of plane circles were the “indivisibles”
of a pyramid, cone, etc.

5. In Cavalieri’s treatment, a moving point
generated a line; a moving line generated a surface; and a
moving surface generated a solid. Cavalieri was not the
first person to consider geometric figures in terms of the
infinitesimal. It had already been incorporated into
medieval scholastic philosophy. Cavalieri was not the
first to use such a concept in computing areas and
volumes, but he was the first one who published a work
on the concept. His work provided a deeper notion of
sets, namely that it isn’t necessary to assign elements to a
set; it is enough that there exists some precise criterion
for determining if an element does or does not belong to
a set.

6. Directorium generale
uranometricum (A General Directory of Uranometry, 1632).
 first to recognize and popularize the value of logarithms in his Directorium generale
uranometricum (A General Directory of Uranometry, 1632).

7. Uranometry is the science of the
measurement of the positions, magnitudes, , etc.
of the stars. The tables of logarithms that he
published included logarithms of trigonometric
functions for use by astronomers. However, his
greatest
was his principle of indivisibles – a procedure that
further developed Archimedes’ method of
exhaustion, although Cavalieri would not be
aware of Archimedes’ work. Cavalieri announced
the principle in 1629, but it did not appear in
print until six years later in his treatise
Geometria indivisibilibus continuorum.

8.  Cavalieri's first book was Lo Specchio Ustorio, overo,
Trattato delle settioni coniche, or The Burning
Mirror, or a Treatise on Conic Sections.[3] In this
book he developed the theory of mirrors shaped
into parabolas, hyperbolas, and ellipses, and various
combinations of these mirrors. The work was purely
theoretical since the needed mirrors could not be
constructed with the technologies of the time, a
limitation well understood by Cavalieri.[4]

9. . One example will suffice. Suppose it be required to find the area of a right-angled
triangle. Let the base be made up of, or contain n points (or indivisibles), and similarly let
the other side contain na points, then the ordinates at the successive points of the base
will contain a, 2a ..., na points. Therefore the number of points in the area is a + 2a + ...
+ na; the sum of which is ½ n²a + ½na. Since n is very large, we may neglect ½na for it is
inconsiderable compared with ½n²a. Hence the area is equal to ½(na)n, that is, ½ ×
altitude × base. There is no difficulty in criticizing such a proof, but, although the form in
which it is presented is indefensible, the substance of it is correct.
It would be misleading to give the above as the only specimen of the method of
indivisibles, and I therefore quote another example, taken from a later writer, which will
fairly illustrate the use of the method when modified and corrected by the method of

10.  Let it be required to find the area outside a parabola APC and
bounded by the curve, the tangent at A, and a line DC parallel
to AB the diameter at A. Complete the parallelogram ABCD.
Divide AD into nequal parts, let AM contain r of them, and
let MN be the (r + 1)th part. Draw MP and NQ parallel to AB, and
draw PR parallel to AD. Then when n becomes indefinitely large,
the curvilinear area APCD will be the the limit of the sum of all
parallelograms like PN.
Now
area PN : area BD = MP.MN : DC.AD.
But by the properties of the parabola
MP : DC = AM² : AD² = r² : n²,
and MN : AD = 1 : n. Hence MP.MN : DC.AD = r² : n³. Therefore
area PN : area BD = r² : n³.
Therefore, ultimately,
area APCD : area BD
= 1² + 2² + ... + (n-1)² : n³
= n (n-1)(2n-1) : n³
which, in the limit, = 1 : 3.

11. Cavalieri’s mathematics -
Integral Calculus
Here is an example of its use.Think about a circle and cutting it up into a large
number of segments which are then stuck down as shown (top to tail):
The shape on the right is ‘almost’ a rectangle and the more segments that are
taken, the closer it will be to a rectangle.
The areas of the circle and the rectangle are the same - they comprise the same
parts. But the length of the rectangle is half the circumference of the circle; the
height of the rectangle is just the radius of the circle. So the area of the circle is
given by the area of the rectangle,
A=(21 2λr).r=λr2

12.  It is perhaps worth noticing that Cavalieri and his
successors always used the method to find the ratio of
two areas, volumes, or magnitudes of the same kind and
dimensions, that is, they never thought of an area as
containing so many units of area. The idea of comparing
a magnitude with a unit of the same kind seems to have
been due to Wallis.
 It is evident that in its direct form the method is
applicable to only a few curves. Cavalieri proved that,
if m be a positive integer, then the limit, when n is
infinite, of is 1/(m+1), which is equivalent to saying
that he found the integral of x to from x = 0 to x = 1; he
also discussed the quadrature of the hyperbola.

13. Geometria indivisibilibus continuorum.
 Geometria indivisibilibus continuorum nova quadam
ratione promota (Geometry, developed by a new method
through the indivisibles of the continua, 1635). In this
work, an area is considered as constituted by an indefinite
number of parallel segments and a volume as constituted
by an indefinite number of parallel planar areas. Such
elements are called indivisibles respectively of area and
volume and provide the building blocks of Cavalieri's
method. As an application, he computed the areas under
the curves – an early integral – which is known
as Cavalieri's quadrature formula.

14. His theory was spurred by Kepler’s Stereometria and
by the encouragement of Galileo. The main
advantage of the method of indivisibles was that it
was more systematic than the method of exhaustion.
In effect, Cavalieri found a result equivalent to
evaluating the integral:
a
∫ xndx as an + 1/(n + 1)
0

15. Kepler’s Stereometria
Kepler's Nova stereometria doliorum vinariorum
 Kepler reported his results on wine barrels in his 1615 book, Nova
stereometria doliorum vinariorum (New solid geometry of wine
barrels). The word Stereometria is from the Ancient
Greek stereos that means solid or three-dimensional
and metron that means a measure or to measure. Stereometria then
means the art of measuring volumes, or solid
geometry. Doliometry is an old-fashioned word from the
Latin dolium that means a large jar or barrel.
 This book is a systematic work on the calculation of areas and
volumes by infinitesimal techniques. Building on the results of
Archimedes, it focuses on solids of revolution and includes
calculations of exact or approximate volumes of over ninety such
solids (Edwards, p. 102). Today we use integral calculus to solve
these kinds of problems.

16. Cavalieri’s method of indivisibles forms a crude type of
integral calculus in which an area is thought of as
consisting of lines and that a solid’s volume can be
regarded as composed of areas that are indivisible. With
his theory he was able to solve many problems connected
with the quadrature of curves and surfaces, finding of
volumes, and locating centers of mass, all of which were
superseded at cone has 1/3 the volume of the prism or
cylinder of equal base and height. He didn’t actually find
the area of a figure or the volume of a solid as being so
many “square units” or “cubic units;” instead he
determined the ratio between the required area or
volume with that of some other easily calculated area or
volume.

17. Cavalieri did not rigorously develop his theory
of indivisibles, but he did not view this as an
important defect. He was intent on finding some
relatively simple practical method for finding areas
and volumes. He was not concerned with the Zeno’s
puzzling paradoxes.

18. This is usually put in the context of a race between Achilles
(the legendary Greek warrior) and the Tortoise. Achilles gives
the Tortoise a head start of, say 10 m, since he runs at 10 ms-
1 and the Tortoise moves at only 1 ms-1. Then by the time
Achilles has reached the point where the Tortoise started (T0 =
10 m), the slow but steady individual will have moved on 1 m to
T1 = 11 m. When Achilles reaches T1, the labouring Tortoise will
have moved on 0.1 m (to T2 = 11.1 m). When Achilles reaches
T2, the Tortoise will still be ahead by 0.01 m, and so on. Each
time Achilles reaches the point where the Tortoise was, the
cunning reptile will always have moved a little way ahead.
The paradoxes of the
philosopher Zeno, born
approximately 490 BC in
southern Italy, have
puzzled mathematicians,
scientists and philosophers
for millennia. Although
none of his work survives
today, over 40 paradoxes
are attributed to him which
appeared in a book he
wrote as a defense of the
philosophies of his teacher
Parmenides. Parmenides
believed in monism, that
reality was a single,
constant, unchanging thing
that he called 'Being'. In
defending this radical
belief, Zeno fashioned 40
arguments to show that
change (motion) and
plurality are impossible.

19.  haunted inquiries into infinite processes.
 He and other mathematicians of the period ignored the
logical imperfections in his use of infinitesimal
quantities. They developed methods whereby whenever a
quantity changed in value according to some continuous
law, as most things in nature seemed to do, the rate of
increase or decrease in such a change was measurable.
Later, when these logical imperfections were removed,
mathematicians developed infinitesimal calculus,
enabling scientists to pry loose the secrets of nature that
for so long had been a closed book.

20. Sir George Gabriel
Stokes, First Baronet,

21. , physicist and mathematician, was born on 13 August 1819 in Skreen, County
Sligo, Ireland. He was the youngest of eight children born to the rector of Skreen,
Gabriel Stokes (1762–1834), and Elizabeth Haughton, the daughter of John
Haugton, the rector of Kilrea, County Londonderry.
was an Irish mathematician and physicist who made many important
contributions to fluid dynamics, optics, and mathematical physics. Together
with James Clerk Maxwell and Lord Kelvin, he was a major contributor to the
fame of the Cambridge school of mathematical physics during the mid-
nineteenth century.
Stoles exerted unusual influence beyond his direct students through extending
assistance in understanding and applying mathematics to any member of the
university. He served in many administrative positions, including for many years
as secretary of the Royal Society. He held strong religious convictions and
published a volume on Natural Theology.

22.  George Gabriel Stokes was the youngest of eight children of the
Reverend Gabriel Stokes, rector of Skreen, County Sligo, and
Elizabeth Haughton. Stokes was raised in an evangelical Protestant
home.
 Stokes was first tutored by a church clerk, but at the age of 13 was
sent to a school in Dublin for a more formal course of education.
Stokes's father died in 1834, but his mother secured the financing to
send him to Bristol College. His mathematics teacher there was
Francis Newman, the brother of Cardinal Newman.
 In 1837, Stokes transferred as an undergraduate to Pembroke
College at the University of Cambridge, where his brother William,
breaking with the family tradition of attending Trinity, had studied.
On graduating as "senior wrangler" and first Smith's prizeman in
1841, Stokes was elected to a fellowship at the college.

23.  THE MOTION OF LIGHT
In physics, the Navier–Stokes equations, named after Claude-Louis
Navier and George Gabriel Stokes, describe the motion of fluid substances. These
equations arise from applying Newton's second law to fluid motion, together with the
assumption that the stress in the fluid is the sum of adiffusing viscous term
(proportional to the gradient of velocity) and a pressure term - hence describing viscous
flow.
The equations are useful because they describe the physics of many things of academic
and economic interest. They may be used to model theweather, ocean currents,
water flow in a pipe and air flow around a wing. The Navier–Stokes equations in their
full and simplified forms help with the design of aircraft and cars, the study of blood
flow, the design of power stations, the analysis of pollution, and many other things.
Coupled withMaxwell's equations they can be used to model and
study magnetohydrodynamics.

24.  Properties of light
Perhaps his best-known researches
are those that deal with the wave
theory of light. His optical work
began at an early period in his
scientific career. His first papers
on the aberration of light
appeared in 1845 and 1846, and
were followed in 1848 by one on
the theory of certain bands seen
in the spectrum. In 1849, he
published a long paper on the
dynamical theory of diffraction,
in which he showed that the
plane of polarization must be
perpendicular to the direction of
propagation.

25.  Fluorescence
In the early 1850s, Stokes
began experimenting with filtered
light. He passed sunlight through a
blue-tinted glass, and then shone
the beam through a solution of
quinone, which has a yellow color.
When the blue light reached the
quinone solution, it produced a
strong yellow illumination. Stokes
tried the same experiment with the
solutions of different compounds,
but found that only some showed
an illumination of a color different
from that of the original light
beam. Stokes named this
effect fluorescence.
 The phenomenon of fluorescence was
known by the middle of the
nineteenth century. British scientist
Sir George G. Stokes first made the
observation that the
mineral fluorspar exhibits
fluorescence when illuminated with
ultraviolet light, and he coined the
word "fluorescence". Stokes observed
that the fluorescing light has longer
wavelengths than the excitation light,
a phenomenon that has become to be
known as the Stokes shift. In Figure
1, a photon of ultraviolet radiation
(purple) collides with an electron in a
simple atom, exciting and elevating
the electron to a higher energy level.
Subsequently, the excited electron
relaxes to a lower level and emits light
in the form of a lower-energy photon
(red) in the visible light region.

26. SPECTROSCOPY
Spectroscopy /spɛkˈtrɒskəpi/ is the
study of the interaction
between matter and radiated
energy.Historically, spectroscopy
originated through the study of visible
light dispersed according to
its wavelength, e.g., by a prism. Later
the concept was expanded greatly to
comprise any interaction with radiative
energy as a function of its wavelength
or frequency. Spectroscopic data is
often represented by a spectrum, a plot
of the response of interest as a function
An excellent example is his work in the theory of
spectroscopy. In his presidential address to the
British Association in 1871, Lord Kelvin (Sir
William Thomson, as he was known then) stated
his belief that the application of the prismatic
analysis of light to solar and stellar chemistry had
never been suggested directly or indirectly by
anyone else when Stokes taught it to him in
Cambridge some time prior to the summer of
1852, and he set forth the conclusions,
theoretical and practical, which he had learned
from Stokes at that time, and which he
afterwards gave regularly in his public lectures at
Glasgow.
These statements, containing as they do the
physical basis on which spectroscopy rests,
and the way in which it is applicable to the
identification of substances existing in
the Sun and stars, make it appear that
Stokes anticipated Kirchhoff by at least seven
or eight years.