Hello, and welcome to Western Sieve Episode two hundred and sixty two, The
Science of Math. Last time we talked about the Earth and the heavens.
We started to get into astronomy, but the Scientific Revolution addressed so many more
aspects of life, and fairly early on, as we're going to see today,
the Scientific Revolution had a profound impact on the field of mathematics. Now,
mathematics was not its own discipline in the year fourteen fifty. It was
considered to be a part of philosophy. Imagine that today you sign up to
major in philosophy fee, and then you're required to take an advanced calculus class.
I can imagine some philosophy majors I know in college who would have been
very annoyed slash terrified at that idea. The Scientific Revolution is going to free
mathematics from philosophy without that first severing, I don't think we ever get to
calculus, or we likely don't. So today we digest changes that were wrought
in the field of the mathematics by the Scientific Revolution. Double entry bookkeeping goes
back to at least the thirteenth century. The principle of double entry is simple.
Every transaction is entered twice as a credit and a debt. So if
I buy a bar of gold worth let's say five hundred dollars, I credit
five hundred to my or an account, and I debit five hundred to my
list of assets. If I borrow five hundred, then five hundred is a
debt to my current account and a credit to my list of liabilities. In
the Renaissance, the standard system involved three books. First, there was a
waste book, in which it recorded everything exactly as it happened, in as
much detail as possible. You could refer to this in the event of a
future dispute or confusion. Then there was a register, in which you turned
your record into a list of transaction. Then the account book proper, with
debts and credits on facing pages. If you check the account book against the
register and the debts against the credits, then you could be confident that the
books were accurate, and every time you balanced the books, you could establish
whether you were making money or losing money. Accounting thus became the basis for
rational investment choices and made it possible to decide how to divide up the profits
of a partnership. Teaching bookkeeping was one of the main ways by which Italian
mathematicians actually earned a living. There was a whole school for this. Now
there seems to be no connection, of course, between bookkeeping and science.
But Galileo, who probably could have taught bookkeeping himself when he was scrabbling a
living together in the years between fifteen eighty five when he ceased to be a
university student, and in fifteen eighty nine when he obtained his first university appointment.
When people complained to Galileo that his law of falling did not correspond to
the real world, because falling objects do not accelerate continuously since they are held
back by air resistance, he would reply that there was simply no contradiction between
the world of theory and the real world, because, to quote him now,
what happens in the concrete happens the same way in the abstract. It
would be novel. Indeed, if computations and ratios made in abstract numbers should
not thereafter correspond to concrete gold and silver, coins, and merchandise. Just
as the bookkeeper who wants his calculations to deal with sugar, silk, and
wool must discount the boxes, bales and other packings so the mathematical scientist,
when he wants to organize in the concrete the effects which he has proven in
the abstract, must adduct the material hindrances, And if he is able to
do so, I assure you that things are in no less agreement than arithmetical
computations. The error is then lie not in the abstract or concreteness, not
in the geometry or physics, but in a calculator who does not how how
to make a true accounting. Double entry bookkeeping thus represented an attempt to make
the real world, the world of bolts of silk and bales of wool and
bags of sugar, mathematically legible. The process of abstraction, it teaches,
is an essential precondition for this new science. In Galileo's day, the other
main source of mathematics in Europe was in painting, specifically in the principles of
perspective representation. Perspective painting was a more recent invention than double entry bookkeeping.
It began between fourteen oh one and fourteen thirteen with Philippio Bruna Skelli. Bruna
Skelly had learned that perspective drawing required establishing a picture plan through which the scene
is viewed, then the artist creates an image that corresponds to how it would
appear if a piece of glass was placed over the plane. There needed to
be a point of perspective and eventually a vanishing point. All of this required
math. Bruna Skelly had learned some thing of enormous importance. For perspective painting
to work, the artist and the viewer had to have their eye located in
the same place. While this wasn't a vanishing point yet per se, European
painters, for the first time in history were on their way. Roughly two
decades separate Bruna Skelly's first studies and the real first large scale painting which fully
masters the technique of perspective representation. This is Macchiocho's famous painting of the Trinity,
created in fourteen twenty five. Massacciotto painting shows Christ on the Cross in
front of a chapel with a barrel vault, but of course the chapel does
not exist. It is entirely a painted chapel. Here's the difference between Bruna
Skelly's studies and Masaccio's painting. Bruna SkELL was representing reality. Massachoko is representing
an imaginary space. You can use the various picture playing techniques to paint reality,
but if you want to paint an imaginary world, then you have to
work out how to construct that world so that it appears convincing and esthetically satisfying.
You have to decide where you want the vanishing point and or the distance
points to be. You have to sketch out a grid of conveying lines.
You have to apply the principles of geometry, and this is exactly what Mascochio
did. We can see the lines he scored in the plaster on which he
painted. We know that Brunu Skelly discussed perspective with Mascacchio, and soon another
artist, Alberti, would write a textbook on geometric perspective. Perspective painting involves
the application of theory to particular circumstances. It trains the eye to think in
terms of geometric shapes. By the middle of the fifteenth century, artists had
begun to think about shapes differently. There could be finite, infinite, abstract,
and undifferentiated space. This is truly the advent of the use of space
in art. Today, types of space and art include positive space, negative
space, deep in, shallow space, and three dimensional space. Positive space
refers to objects that stand out from the negative surrounding or background. Deep space
refers to the depth, and shallow space refers to the lack of depth.
All of these ideas get their beginning in the fifteenth century, and they are
essentially math. It's hard to express how important this innovation was for the purpose
of invention. Before perspective drawing, if you want to design a piece of
machinery, you had to make it or a model of it. But once
they had perspective drawing, you could design with only a pencil itself. Invented
in fifteen sixty, da Vinci is a great exemplar here. He designed hundreds
of contraptions that were never built and frankly, could never be built. But
the fact that his ideas could not be converted into a physical model no longer
held him back from dreaming and inventing. Perspective painting made the impossible possible.
The other major revolution to human kinds understanding of the world was the engraved plate,
first used in fifteen forty three. Engraved plates allowed printers to easily,
relatively speaking, put images into books. Images that used perspective. Thus,
by fifteen forty three, two revolutions come together to make a new type of
science. On the one hand, there was perspective painting rounded in geometrical abstraction.
On the other, the printing of engraved plates supplemented by text produced on
a printing press. Perspective painting goes back to fourteen twenty five, engraved prints
to at least fourteen twenty eight, the printing press to fourteen fifty. The
fall of Constantinople, one consequence of which was a flood of Greek manuscripts and
Greek speaking scholars entering the Latin speaking West from the east, occurred in fourteen
fifty three. Why then, did it take a further century to complete the
transformation brought about by the mechanical reproduction of perspective images. There's two answers to
this question. First, the immediate priority of publishers in the years after the
invention of printing was to publish the vast body of religious, philosophical, and
literary texts which had been inherited from the past, first the Latin texts,
and then from more limited audience, the Greek ones. The first reliable addition
of Galen on which Visalius had worked, appeared in Basil in fifteen thirty eight.
Second, a long cultural revolution still had to take place, in which
book learning came to seem of lesser importance than direct experience. That revolution began
with Columbus. In fourteen sixty four, a German astronomer, Johannes Mueller,
known as Reggio Montanus. Reggio Montanas was a version of the place he came
for, Klnisberg in Latin, gave a lecture at the University of Padua.
Reggio Montanus had recently completed an exposition and commentary on Ptolemay's astronomy, begun by
his mentor, George Purebuck. This was to become the standard textbook in advanced
astronomy for the whole of the sixteenth century, and in its Purebach and Reggiomontanus
did not hesitate to criticize Ptolemay for his errors. In fourteen sixty four,
Reggio Montanas was writing on a path breaking guide to plain and spherical trigonometry,
which laid out the mathematical foundations for astronomical calculations. He had learned Greek in
Vienna in order to read Ptolemay in the original and in Italy. He had
been able to read in Greek Archimedes, who had been translated into Latin in
the Middle Ages but was not yet available in print. Reggio Montanus was the
first to really benefit from the supply of ancient Greek texts that reached Italy after
the fall of Constantinople. At the time of his lecture in Padua, less
than a decade after the publication of the Gutenberg Bible, the printing revolution was
only just beginning to get under way. Euclid, for example, was first
printed in Latin in fourteen eighty four, in Greek in fifteen thirty three,
and finally in Italian in fifteen forty three, in English in fifteen seventy.
Reggio Montanus's lecture thus marks a key moment in the re acquisition of Greek mathematics
and points towards the ambitious program for the publication of mathematical texts that Reggio Montanus
developed, though he died, as I'll talk about more in later episodes before
it could be carried out. Reggio Montanus also spoke in praise of the mathematical
sciences, and he praised them by Denacra. Aristotelian philosophy taught new universities.
Even Aristotle, he wrote, if he came back to life, would not
be able to make sense of what was said by his modern disciples. In
fourteen seventy one, Reggiomontanus worked out a procedure for measuring the parallax of heavenly
bodies and so their distance from the earth. His procedure presumed the use of
what was called a cross staff, which was an instrument invented by a Jewish
rabbi, Levi Ben Gerson in thirteen twenty eight. It's a very simple instrument.
Essentially, it's just a calibrated shaft along which a bar slides. You
cite the shaft and move the bar back and forth until you've lined up its
ends with two points, and the angle can then be read off the scale
from the shaft. You can use a cross staff, for example, to
measure the angle between the horizon and the sun at mid day. If you
know the date and you have the right tables, you can then read off
your la stude. This of course involves squinting at the sun, which a
lot of pilots in the early age of the age of discovery wind up going
blind from Alternatively, at night, you could measure latitude directly by measuring the
angle between the horizon and the pulstar. The cross staff is merely one of
a series of instruments, such as the quadrant or sextant, designed for measuring
angles by taking sightings. Before it was invented, the Astra lab had provided
a sighting device and also a method for measuring the height of the sun from
its shadow. Again, with this device, you can establish your latitude if
you know your time of day. But more importantly, for most uters,
you could tell the time of day if you know your latitude and the date.
Specialist forms of all these instruments were developed for surveying, for astronomy,
for navigation, but the basic principle that angles could be used to determine distances
or times was the same for all of them. So what we're seeing in
the early part of the scientific revolution is for the first time, math is
getting consistent and consistently used amongst different practitioners in surveying. If you know how
far away a building was, it was now easy to calculate its height.
Suppose you wanted to scale the walls of a fortress which were on the other
side of a river. You could take two measurements in a straight line with
the building, and from the distance between the measurements and the difference between the
angles as measured with a cross staff, you could calculate the height of the
walls and make your ladders to the right height. The basic principles involved had
been described by Euclid and we're well understood in the Middle Ages. They are
exactly the same principles as are involved in perspective painting. But where perspective painting
takes a three dimensional world and turns it into a two dimensional surface, Reggio
Montanas was now trying to take a two dimensional image the night sky and turn
it into a three dimensional world. To do so, you have to,
in effect moved from monocular vision to binocular vision. So the principle of the
parallax is what enables mathematicians to do this. It's a variation of the basic
principle that if you know one angle and one side of an equal adteral or
right sided triangle, you can determine the other angles and sides. It thus
requires not one measurement but two. For example, hold your finger up in
front of you. Not if your driving, close your left eye again.
Not. If you're driving and know where your finger appears to be against the
background, then switch eyes immediately. Your finger is going to jump to the
right. If you know your distance between your eyes and measure the angle that
corresponds to the apparent shift in your finger's position, you can calculate exactly how
far away your finger is, although of course no one's going to do that.
In this case, the distance between your eye is significant portion of the
distance between your eyes and your fingers. If you're trying to measure the distance
to an object that was very far away, you would have to set up
two observation spots that were far apart, or at least would seem to be.
Reggio Montanus grasped that an astronomer does not have to travel in order to
get two observation points that are in effect far apart. If the heavens rotate
around the center of the universe, and if that center is at or near
the center of the Earth, then the observation point of the astronomer who is
on the surface of the Earth changes in its relationship to the heavens as they
move. Simply because the astronomer is not looking at the heavens from the center
of the universe, but from a point that is distant from the center.
So imagine you're standing at the immediate dead center of a merry go round or
a carousel on which horses are arranged in concentric circles. At the center is
a stationary round platform, around which the circle of horses, each taking the
same time to complete a circuit, revolve. As you look outwards and the
horses turn around you, the relative position of the horses will remain the same.
A horse which is in line with another horse at one moment will still
be in line with it a quarter of a revolution later. But if you
take a few steps in any direction until you reach the edge of the stationary
platform, then the relative position of the horses will appear to change all the
time perspective changes. Moreover, if you know the size of the stationary platform
the distance to the outer ring of the horses, then you can use changes
in relative position of the horses in the two other rings to work out how
far away they are. In essence, Reggio Montanas figured out that you could
measure the parallax of heavenly bodies by taking two observations from the same place but
at different times, rather than taking two observations from different places but at the
same time. Although Reggio Montanas worked out how to make such a measurement in
fourteen seventy one, the full account of this procedure didn't get published until fifteen
thirty one. Unfortunately, in fifteen forty eight, a text apparently by Reggio
Montanas was published which claimed to measure the parallax of the comet which had appeared
in fourteen seventy two, and to confirm that it was as close to the
Earth because the paradox was a whopping six degrees, placing it much closer than
the moon, which had a diurnal parallax about one degree. Now sadly,
recent historians have established that this particular paper was not written by Reggio Montanas.
It was found amongst his papers when he died. It was presumably in his
handwriting. However, he didn't write it. He might have been working on
it, he might have been copying something on it. But we'd know that.
No one in the sixteenth century realized this, and that turned out to
be the cause of a lot of confusion then in fifteen seventy two, the
world changed again. Astronomer Tico Brahey noticed a new star in the sky.
For a time, it was the brightest object in the heavens other than the
Sun and the moon, brighter even than Venus. Such events only occur once
in a thousand years or so, and unlike a comet, the new star
stood still, which made it much easier to measure its parallax. All over
Europe astronomers were literally obsessed with it. Since they now knew Reggio Montanus's real
technique for measuring parallax, they naturally tried to apply it. Some found a
measurable parallax, but others insisted that they couldn't it didn't exist accurately. Measuring
parallax from far was easy, particularly as it required a more exact measurement of
time than a sixteenth century clock could provide, but showing that there was no
measurable parallax was much more straightforward. All one had to do was hold up
a thread as a sighting device and find two stars that were exactly in line
with this new one, but north or south of it. If the same
stars were exactly in line with the nova later that same night, then there
was no parallax. This simple technique was actually employed by the teacher of Johannes
Kepler later on. And if there was no parallax, then the comet must
be a vast distance away, far further than the Moon, whose parallax was
quite easy to measure, and it must be super luminary, not a sublunary
body. But how to explain the appearance of a new star in the heavens?
Since there could be no natural explanation assuming the star was indeed the heavens,
the event was just a miracle sent by God. The finest astronomers and
astrologers, including Thomas Diggs in England, racked their brains in an attempt to
figure out what it might portend, and hastened to publish their conflicting conclusions,
making matters more complicated. The new star of fifteen seventy two was followed by
a new Comment of fifteen seventy seven, and here again parallax measurement placed the
comment far beyond the moon, where a nova or a new star could possibly
regard it as a miracle. A comment was too commonplace to be handled in
that way. Ra Hay worked out a further problem that could be solved by
measuring parallax. A crucial difference between the Ptolemaic system on the one hand and
the Copernican system on the other was that under these modern systems, Mars must
at all times approach much closer to the Earth than under the Ptolemaic system.
Ra Hay at first thought he had obtained a reliable figure for the parallax of
Mars, which proved the Ptolemaic system was mistaken, although he later realized there
were problems with that Reggio Montana. This procedure for measuring parallax ideally involved comparing
the apparent position of a celestial object soon after dark with its apparent position not
long before dawn, thus maximizing the parallax to be measured. Neither the New
Star of fifteen seventy two nor the Comment of fifteen seventy seven set in the
night sky viewed from northern Europe, so the ideal procedure was inapplicable. In
the case of Mars. There was no choice but to make measurements when the
planet was nearly in line with the Sun, and thus it never rose high
above the horizon at night. In measuring the location of an object near the
horizon, ray had to allow for the refraction caused by the greater thickness of
the atmosphere which with its rays had passed, and eventually what he found was
that he had miscalculated this. But as was the case in many times in
the scientific Revolution, what mattered not was that the individual was wrong. The
fact that tco Brian was wrong about Mars didn't make a difference. What mattered
was his method was right, and the observations that he took day after day,
night after night, proved instrumental for later astronomers who are able to correct
the error. Again, this all gets down to whether or not Europeans are
allowed to question knowledge under the ancient classical system. The answer is absolutely not.
And so even if you don't understand any of this stuff about parallaxes and
so on and so forth, and I only loosely grasp it, what you
should come away with this is understanding that for the first time people were willing
to take math and try to apply it in new, in unique ways to
the world, in a way in which our understanding of the natural world became
so much more systematized, so much easier to measure, and this is a
great example of a few fundamental features of the scientific Revolution. The first is
inevitability. Once Reggio Montanas developed a system for measuring parallax and published it,
astronomers were set on a path that could only lead to proving Aristotle and Ptolemy
were wrong. The second feature is time. Delays in publication meant that it
took a long time for the inevitable conclusions to hit home, but ultimately that
would not matter because of a shift in the mindset from Europeans from theorizing and
logical argument to experimentation and objective proofs. Once a discovery had been made,
there was no one making it, not that some didn't try. We'll get
into the inquisition here in a few weeks when I discuss reactions to the scientific
Revolution. And while I do not want to overly belabor the point, the
second fundamental feature of the scientific revolution remains the impact of the printing press.
Printing made it possible for Ray to survey a wide range of publications. Before
he turned his gaze to the skies, he looked to a book, well
multiple books, none of which would have been possible without printing. Historians of
science have often and rightly suggested that the key to the scientific revolution is the
math the midization of nature. Aristotle and Ptolema had assumed that the heavens were
mathematically legible, and indeed Ptolemy had devised techniques for reading them. One aspect
of the scientific revolution consists in the extension of mathematical theories to include sublunary phenomena.
Where Aristotelian physics was preoccupied with qualities the four elements earth, air,
fire, and water, and embody the four qualities hot, dry, cold,
and wet. You'll remember that from the four humors, the new physics
was preoccupied with movements and quantities that could be measured, and it quickly led
to attempts to measure the speed of falling bodies, the speed of sound,
and the weight of air. Where Aristotle had assumed that each element behaved differently,
the new physics assumed that all heavy objects could be thought of as the
same. Where Aristotelian physics had depended on all five senses, the new physics
relied only on site. While traditional histories suggest that the mathematization of science started
in the seventeenth century with new physics, really, we do get a glimpse
of it much earlier through perspective painting. Galileo learned math from Hostilio Ricci,
who by trade was a teacher of perspective to artists, not astronomers. The
amazing thing is how an innovation in one field can now jump to another.
It can lead to innovations in other, seemingly unconnected branches of science. In
the sixteenth century, perspective and coordinators made the jump from perspective painting to geography.
Now there wasn't necessarily anything new about this. Cicero had thought that geography
was a branch of geometry. With geometry came abstraction. Perspective is represented with
two parallel lines converging towards a vanishing point. It's in fact a grid used
by artists to establish a picture plane, and the exact same thing can be
done to establish latitude and longitude. Geometry, by the way, also acquired
new importance as a result of the invention of gunpowder. Fortifications had to be
built to resist cannonballs, which fly in straight lines at least as a bird
season. In order to provide raking and flanking fire along every wall, forts
needed to be designed on the page with very carefully measured angles and distances.
Thus, if we were to ask how the scientific revolution became mathematized, the
answer is clear. Perspective painting led to cartography, which led to navigation and
proper astronomy, and ultimately to ballistics. And in many cases all of these
changes in innovations happened simultaneously, with one technique jumping seamlessly from one branch of
science to the next. These were like real things too, that made a
difference in the actual world. In sixteen twenty two, for example, a
fleet of Dutch ships tried to seize the Portuguese colony of Macao. A Jesuit
mathematician did the geometry calculations to determine the distance to a stockpile of gunpowder that
Dutch had brought ashore, and the angle of elevation at which the cannon should
be set. A direct hit turned the tide of that battle and ensured that
Macau remain a Portuguese colony. Bus If we ask how did the scientific revolution
become mathematized, the answer is through the different fields that I've already laid out.
And of course, and again I don't want to keep the laboring this
point, but the answer is also because of printing, because printing allowed the
spread of ideas from one branch of science to the next, and often within
the same. Frankly, first and foremost, we might want to think about
the scientific revolution as a revolt, a revolt by the mathematicians against the authority
of philosophers, an authority they had held for thousands of years. We'll end
it there for today, before continuing with the scientific Revolution next week. As
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