WEBVTT 1 00:00:16.800 --> 00:00:22.519 Hello, and welcome to Western Sieve Episode two hundred and sixty two, The 2 00:00:22.640 --> 00:00:28.480 Science of Math. Last time we talked about the Earth and the heavens. 3 00:00:29.440 --> 00:00:35.560 We started to get into astronomy, but the Scientific Revolution addressed so many more 4 00:00:35.600 --> 00:00:41.079 aspects of life, and fairly early on, as we're going to see today, 5 00:00:41.439 --> 00:00:48.079 the Scientific Revolution had a profound impact on the field of mathematics. Now, 6 00:00:48.159 --> 00:00:53.119 mathematics was not its own discipline in the year fourteen fifty. It was 7 00:00:53.159 --> 00:00:58.640 considered to be a part of philosophy. Imagine that today you sign up to 8 00:00:58.840 --> 00:01:03.560 major in philosophy fee, and then you're required to take an advanced calculus class. 9 00:01:03.840 --> 00:01:07.680 I can imagine some philosophy majors I know in college who would have been 10 00:01:07.760 --> 00:01:15.879 very annoyed slash terrified at that idea. The Scientific Revolution is going to free 11 00:01:15.920 --> 00:01:22.319 mathematics from philosophy without that first severing, I don't think we ever get to 12 00:01:22.359 --> 00:01:30.280 calculus, or we likely don't. So today we digest changes that were wrought 13 00:01:30.280 --> 00:01:42.959 in the field of the mathematics by the Scientific Revolution. Double entry bookkeeping goes 14 00:01:44.000 --> 00:01:49.280 back to at least the thirteenth century. The principle of double entry is simple. 15 00:01:49.040 --> 00:01:53.560 Every transaction is entered twice as a credit and a debt. So if 16 00:01:53.599 --> 00:01:59.120 I buy a bar of gold worth let's say five hundred dollars, I credit 17 00:01:59.159 --> 00:02:01.879 five hundred to my or an account, and I debit five hundred to my 18 00:02:01.920 --> 00:02:07.719 list of assets. If I borrow five hundred, then five hundred is a 19 00:02:07.760 --> 00:02:12.520 debt to my current account and a credit to my list of liabilities. In 20 00:02:12.560 --> 00:02:16.240 the Renaissance, the standard system involved three books. First, there was a 21 00:02:16.280 --> 00:02:21.479 waste book, in which it recorded everything exactly as it happened, in as 22 00:02:21.520 --> 00:02:24.599 much detail as possible. You could refer to this in the event of a 23 00:02:24.599 --> 00:02:30.560 future dispute or confusion. Then there was a register, in which you turned 24 00:02:30.560 --> 00:02:35.479 your record into a list of transaction. Then the account book proper, with 25 00:02:35.560 --> 00:02:39.240 debts and credits on facing pages. If you check the account book against the 26 00:02:39.280 --> 00:02:44.439 register and the debts against the credits, then you could be confident that the 27 00:02:44.439 --> 00:02:47.879 books were accurate, and every time you balanced the books, you could establish 28 00:02:47.919 --> 00:02:53.639 whether you were making money or losing money. Accounting thus became the basis for 29 00:02:53.800 --> 00:02:58.840 rational investment choices and made it possible to decide how to divide up the profits 30 00:02:59.000 --> 00:03:04.759 of a partnership. Teaching bookkeeping was one of the main ways by which Italian 31 00:03:04.800 --> 00:03:09.400 mathematicians actually earned a living. There was a whole school for this. Now 32 00:03:09.479 --> 00:03:14.400 there seems to be no connection, of course, between bookkeeping and science. 33 00:03:15.319 --> 00:03:20.360 But Galileo, who probably could have taught bookkeeping himself when he was scrabbling a 34 00:03:20.400 --> 00:03:23.560 living together in the years between fifteen eighty five when he ceased to be a 35 00:03:23.639 --> 00:03:29.759 university student, and in fifteen eighty nine when he obtained his first university appointment. 36 00:03:30.120 --> 00:03:35.919 When people complained to Galileo that his law of falling did not correspond to 37 00:03:35.960 --> 00:03:40.199 the real world, because falling objects do not accelerate continuously since they are held 38 00:03:40.240 --> 00:03:46.400 back by air resistance, he would reply that there was simply no contradiction between 39 00:03:46.400 --> 00:03:51.479 the world of theory and the real world, because, to quote him now, 40 00:03:52.280 --> 00:03:55.560 what happens in the concrete happens the same way in the abstract. It 41 00:03:55.599 --> 00:04:01.039 would be novel. Indeed, if computations and ratios made in abstract numbers should 42 00:04:01.039 --> 00:04:06.759 not thereafter correspond to concrete gold and silver, coins, and merchandise. Just 43 00:04:06.840 --> 00:04:12.240 as the bookkeeper who wants his calculations to deal with sugar, silk, and 44 00:04:12.400 --> 00:04:18.199 wool must discount the boxes, bales and other packings so the mathematical scientist, 45 00:04:18.240 --> 00:04:23.439 when he wants to organize in the concrete the effects which he has proven in 46 00:04:23.480 --> 00:04:28.240 the abstract, must adduct the material hindrances, And if he is able to 47 00:04:28.279 --> 00:04:32.519 do so, I assure you that things are in no less agreement than arithmetical 48 00:04:32.720 --> 00:04:39.879 computations. The error is then lie not in the abstract or concreteness, not 49 00:04:40.040 --> 00:04:44.920 in the geometry or physics, but in a calculator who does not how how 50 00:04:44.959 --> 00:04:53.279 to make a true accounting. Double entry bookkeeping thus represented an attempt to make 51 00:04:53.480 --> 00:04:58.519 the real world, the world of bolts of silk and bales of wool and 52 00:04:58.600 --> 00:05:03.120 bags of sugar, mathematically legible. The process of abstraction, it teaches, 53 00:05:03.519 --> 00:05:10.439 is an essential precondition for this new science. In Galileo's day, the other 54 00:05:10.480 --> 00:05:15.839 main source of mathematics in Europe was in painting, specifically in the principles of 55 00:05:15.920 --> 00:05:24.399 perspective representation. Perspective painting was a more recent invention than double entry bookkeeping. 56 00:05:25.199 --> 00:05:31.399 It began between fourteen oh one and fourteen thirteen with Philippio Bruna Skelli. Bruna 57 00:05:31.439 --> 00:05:38.600 Skelly had learned that perspective drawing required establishing a picture plan through which the scene 58 00:05:38.639 --> 00:05:44.000 is viewed, then the artist creates an image that corresponds to how it would 59 00:05:44.000 --> 00:05:47.959 appear if a piece of glass was placed over the plane. There needed to 60 00:05:47.959 --> 00:05:56.439 be a point of perspective and eventually a vanishing point. All of this required 61 00:05:56.720 --> 00:06:03.360 math. Bruna Skelly had learned some thing of enormous importance. For perspective painting 62 00:06:03.439 --> 00:06:09.160 to work, the artist and the viewer had to have their eye located in 63 00:06:09.199 --> 00:06:15.800 the same place. While this wasn't a vanishing point yet per se, European 64 00:06:15.879 --> 00:06:21.439 painters, for the first time in history were on their way. Roughly two 65 00:06:21.560 --> 00:06:29.680 decades separate Bruna Skelly's first studies and the real first large scale painting which fully 66 00:06:29.720 --> 00:06:36.360 masters the technique of perspective representation. This is Macchiocho's famous painting of the Trinity, 67 00:06:36.839 --> 00:06:44.720 created in fourteen twenty five. Massacciotto painting shows Christ on the Cross in 68 00:06:44.759 --> 00:06:49.079 front of a chapel with a barrel vault, but of course the chapel does 69 00:06:49.120 --> 00:06:56.000 not exist. It is entirely a painted chapel. Here's the difference between Bruna 70 00:06:56.040 --> 00:07:04.439 Skelly's studies and Masaccio's painting. Bruna SkELL was representing reality. Massachoko is representing 71 00:07:04.720 --> 00:07:12.839 an imaginary space. You can use the various picture playing techniques to paint reality, 72 00:07:13.279 --> 00:07:16.160 but if you want to paint an imaginary world, then you have to 73 00:07:16.160 --> 00:07:24.160 work out how to construct that world so that it appears convincing and esthetically satisfying. 74 00:07:25.480 --> 00:07:30.120 You have to decide where you want the vanishing point and or the distance 75 00:07:30.160 --> 00:07:33.879 points to be. You have to sketch out a grid of conveying lines. 76 00:07:35.120 --> 00:07:42.399 You have to apply the principles of geometry, and this is exactly what Mascochio 77 00:07:42.560 --> 00:07:46.560 did. We can see the lines he scored in the plaster on which he 78 00:07:46.639 --> 00:07:54.639 painted. We know that Brunu Skelly discussed perspective with Mascacchio, and soon another 79 00:07:54.800 --> 00:08:01.959 artist, Alberti, would write a textbook on geometric perspective. Perspective painting involves 80 00:08:01.000 --> 00:08:07.680 the application of theory to particular circumstances. It trains the eye to think in 81 00:08:07.759 --> 00:08:13.279 terms of geometric shapes. By the middle of the fifteenth century, artists had 82 00:08:13.360 --> 00:08:18.680 begun to think about shapes differently. There could be finite, infinite, abstract, 83 00:08:18.959 --> 00:08:26.120 and undifferentiated space. This is truly the advent of the use of space 84 00:08:26.399 --> 00:08:31.080 in art. Today, types of space and art include positive space, negative 85 00:08:31.120 --> 00:08:37.159 space, deep in, shallow space, and three dimensional space. Positive space 86 00:08:37.200 --> 00:08:43.879 refers to objects that stand out from the negative surrounding or background. Deep space 87 00:08:43.919 --> 00:08:48.399 refers to the depth, and shallow space refers to the lack of depth. 88 00:08:48.799 --> 00:08:54.519 All of these ideas get their beginning in the fifteenth century, and they are 89 00:08:54.759 --> 00:09:03.559 essentially math. It's hard to express how important this innovation was for the purpose 90 00:09:03.879 --> 00:09:11.039 of invention. Before perspective drawing, if you want to design a piece of 91 00:09:11.080 --> 00:09:16.840 machinery, you had to make it or a model of it. But once 92 00:09:16.919 --> 00:09:22.759 they had perspective drawing, you could design with only a pencil itself. Invented 93 00:09:24.039 --> 00:09:31.480 in fifteen sixty, da Vinci is a great exemplar here. He designed hundreds 94 00:09:31.519 --> 00:09:37.879 of contraptions that were never built and frankly, could never be built. But 95 00:09:37.919 --> 00:09:43.960 the fact that his ideas could not be converted into a physical model no longer 96 00:09:43.039 --> 00:09:52.799 held him back from dreaming and inventing. Perspective painting made the impossible possible. 97 00:09:54.120 --> 00:10:01.360 The other major revolution to human kinds understanding of the world was the engraved plate, 98 00:10:01.159 --> 00:10:07.600 first used in fifteen forty three. Engraved plates allowed printers to easily, 99 00:10:07.879 --> 00:10:16.639 relatively speaking, put images into books. Images that used perspective. Thus, 100 00:10:16.639 --> 00:10:20.840 by fifteen forty three, two revolutions come together to make a new type of 101 00:10:20.879 --> 00:10:28.000 science. On the one hand, there was perspective painting rounded in geometrical abstraction. 102 00:10:28.840 --> 00:10:33.399 On the other, the printing of engraved plates supplemented by text produced on 103 00:10:33.399 --> 00:10:39.519 a printing press. Perspective painting goes back to fourteen twenty five, engraved prints 104 00:10:39.759 --> 00:10:45.720 to at least fourteen twenty eight, the printing press to fourteen fifty. The 105 00:10:45.799 --> 00:10:50.519 fall of Constantinople, one consequence of which was a flood of Greek manuscripts and 106 00:10:50.559 --> 00:10:56.519 Greek speaking scholars entering the Latin speaking West from the east, occurred in fourteen 107 00:10:56.600 --> 00:11:00.879 fifty three. Why then, did it take a further century to complete the 108 00:11:00.919 --> 00:11:07.440 transformation brought about by the mechanical reproduction of perspective images. There's two answers to 109 00:11:07.559 --> 00:11:13.200 this question. First, the immediate priority of publishers in the years after the 110 00:11:13.240 --> 00:11:18.960 invention of printing was to publish the vast body of religious, philosophical, and 111 00:11:18.080 --> 00:11:24.720 literary texts which had been inherited from the past, first the Latin texts, 112 00:11:24.759 --> 00:11:30.759 and then from more limited audience, the Greek ones. The first reliable addition 113 00:11:30.840 --> 00:11:35.720 of Galen on which Visalius had worked, appeared in Basil in fifteen thirty eight. 114 00:11:37.840 --> 00:11:43.519 Second, a long cultural revolution still had to take place, in which 115 00:11:43.559 --> 00:11:50.080 book learning came to seem of lesser importance than direct experience. That revolution began 116 00:11:50.279 --> 00:11:54.759 with Columbus. In fourteen sixty four, a German astronomer, Johannes Mueller, 117 00:11:56.120 --> 00:12:01.679 known as Reggio Montanus. Reggio Montanas was a version of the place he came 118 00:12:01.759 --> 00:12:07.240 for, Klnisberg in Latin, gave a lecture at the University of Padua. 119 00:12:07.519 --> 00:12:13.720 Reggio Montanus had recently completed an exposition and commentary on Ptolemay's astronomy, begun by 120 00:12:13.759 --> 00:12:18.720 his mentor, George Purebuck. This was to become the standard textbook in advanced 121 00:12:18.720 --> 00:12:24.440 astronomy for the whole of the sixteenth century, and in its Purebach and Reggiomontanus 122 00:12:24.840 --> 00:12:31.080 did not hesitate to criticize Ptolemay for his errors. In fourteen sixty four, 123 00:12:31.480 --> 00:12:37.000 Reggio Montanas was writing on a path breaking guide to plain and spherical trigonometry, 124 00:12:37.639 --> 00:12:43.200 which laid out the mathematical foundations for astronomical calculations. He had learned Greek in 125 00:12:43.360 --> 00:12:48.559 Vienna in order to read Ptolemay in the original and in Italy. He had 126 00:12:48.559 --> 00:12:52.519 been able to read in Greek Archimedes, who had been translated into Latin in 127 00:12:52.519 --> 00:12:56.639 the Middle Ages but was not yet available in print. Reggio Montanus was the 128 00:12:56.720 --> 00:13:03.600 first to really benefit from the supply of ancient Greek texts that reached Italy after 129 00:13:03.639 --> 00:13:09.879 the fall of Constantinople. At the time of his lecture in Padua, less 130 00:13:09.879 --> 00:13:15.480 than a decade after the publication of the Gutenberg Bible, the printing revolution was 131 00:13:15.519 --> 00:13:20.519 only just beginning to get under way. Euclid, for example, was first 132 00:13:20.559 --> 00:13:26.279 printed in Latin in fourteen eighty four, in Greek in fifteen thirty three, 133 00:13:26.519 --> 00:13:31.120 and finally in Italian in fifteen forty three, in English in fifteen seventy. 134 00:13:33.159 --> 00:13:39.720 Reggio Montanus's lecture thus marks a key moment in the re acquisition of Greek mathematics 135 00:13:39.360 --> 00:13:46.120 and points towards the ambitious program for the publication of mathematical texts that Reggio Montanus 136 00:13:46.159 --> 00:13:50.440 developed, though he died, as I'll talk about more in later episodes before 137 00:13:50.440 --> 00:13:56.360 it could be carried out. Reggio Montanus also spoke in praise of the mathematical 138 00:13:56.399 --> 00:14:01.919 sciences, and he praised them by Denacra. Aristotelian philosophy taught new universities. 139 00:14:03.080 --> 00:14:07.480 Even Aristotle, he wrote, if he came back to life, would not 140 00:14:07.519 --> 00:14:13.120 be able to make sense of what was said by his modern disciples. In 141 00:14:13.159 --> 00:14:18.279 fourteen seventy one, Reggiomontanus worked out a procedure for measuring the parallax of heavenly 142 00:14:18.360 --> 00:14:22.840 bodies and so their distance from the earth. His procedure presumed the use of 143 00:14:22.879 --> 00:14:28.240 what was called a cross staff, which was an instrument invented by a Jewish 144 00:14:28.320 --> 00:14:33.440 rabbi, Levi Ben Gerson in thirteen twenty eight. It's a very simple instrument. 145 00:14:33.840 --> 00:14:39.200 Essentially, it's just a calibrated shaft along which a bar slides. You 146 00:14:39.399 --> 00:14:43.240 cite the shaft and move the bar back and forth until you've lined up its 147 00:14:43.320 --> 00:14:46.440 ends with two points, and the angle can then be read off the scale 148 00:14:46.440 --> 00:14:50.919 from the shaft. You can use a cross staff, for example, to 149 00:14:50.960 --> 00:14:56.600 measure the angle between the horizon and the sun at mid day. If you 150 00:14:56.679 --> 00:14:58.600 know the date and you have the right tables, you can then read off 151 00:14:58.600 --> 00:15:03.639 your la stude. This of course involves squinting at the sun, which a 152 00:15:03.679 --> 00:15:09.799 lot of pilots in the early age of the age of discovery wind up going 153 00:15:09.840 --> 00:15:13.919 blind from Alternatively, at night, you could measure latitude directly by measuring the 154 00:15:13.960 --> 00:15:18.960 angle between the horizon and the pulstar. The cross staff is merely one of 155 00:15:18.960 --> 00:15:24.879 a series of instruments, such as the quadrant or sextant, designed for measuring 156 00:15:24.919 --> 00:15:30.440 angles by taking sightings. Before it was invented, the Astra lab had provided 157 00:15:30.440 --> 00:15:33.960 a sighting device and also a method for measuring the height of the sun from 158 00:15:33.960 --> 00:15:37.480 its shadow. Again, with this device, you can establish your latitude if 159 00:15:37.480 --> 00:15:41.279 you know your time of day. But more importantly, for most uters, 160 00:15:41.759 --> 00:15:46.639 you could tell the time of day if you know your latitude and the date. 161 00:15:46.600 --> 00:15:52.159 Specialist forms of all these instruments were developed for surveying, for astronomy, 162 00:15:52.519 --> 00:15:58.679 for navigation, but the basic principle that angles could be used to determine distances 163 00:15:58.240 --> 00:16:03.200 or times was the same for all of them. So what we're seeing in 164 00:16:03.240 --> 00:16:08.159 the early part of the scientific revolution is for the first time, math is 165 00:16:08.159 --> 00:16:17.039 getting consistent and consistently used amongst different practitioners in surveying. If you know how 166 00:16:17.080 --> 00:16:21.039 far away a building was, it was now easy to calculate its height. 167 00:16:21.879 --> 00:16:25.039 Suppose you wanted to scale the walls of a fortress which were on the other 168 00:16:25.120 --> 00:16:29.559 side of a river. You could take two measurements in a straight line with 169 00:16:29.639 --> 00:16:33.840 the building, and from the distance between the measurements and the difference between the 170 00:16:33.840 --> 00:16:37.519 angles as measured with a cross staff, you could calculate the height of the 171 00:16:37.559 --> 00:16:42.519 walls and make your ladders to the right height. The basic principles involved had 172 00:16:42.559 --> 00:16:48.840 been described by Euclid and we're well understood in the Middle Ages. They are 173 00:16:48.879 --> 00:16:53.639 exactly the same principles as are involved in perspective painting. But where perspective painting 174 00:16:53.919 --> 00:17:00.519 takes a three dimensional world and turns it into a two dimensional surface, Reggio 175 00:17:00.600 --> 00:17:06.480 Montanas was now trying to take a two dimensional image the night sky and turn 176 00:17:06.519 --> 00:17:11.480 it into a three dimensional world. To do so, you have to, 177 00:17:11.720 --> 00:17:18.799 in effect moved from monocular vision to binocular vision. So the principle of the 178 00:17:18.839 --> 00:17:23.799 parallax is what enables mathematicians to do this. It's a variation of the basic 179 00:17:23.839 --> 00:17:29.279 principle that if you know one angle and one side of an equal adteral or 180 00:17:29.359 --> 00:17:33.759 right sided triangle, you can determine the other angles and sides. It thus 181 00:17:33.920 --> 00:17:40.400 requires not one measurement but two. For example, hold your finger up in 182 00:17:40.440 --> 00:17:42.559 front of you. Not if your driving, close your left eye again. 183 00:17:42.640 --> 00:17:47.359 Not. If you're driving and know where your finger appears to be against the 184 00:17:47.359 --> 00:17:52.200 background, then switch eyes immediately. Your finger is going to jump to the 185 00:17:52.279 --> 00:17:57.599 right. If you know your distance between your eyes and measure the angle that 186 00:17:57.640 --> 00:18:03.799 corresponds to the apparent shift in your finger's position, you can calculate exactly how 187 00:18:03.839 --> 00:18:07.599 far away your finger is, although of course no one's going to do that. 188 00:18:08.799 --> 00:18:12.519 In this case, the distance between your eye is significant portion of the 189 00:18:12.559 --> 00:18:18.920 distance between your eyes and your fingers. If you're trying to measure the distance 190 00:18:18.960 --> 00:18:22.640 to an object that was very far away, you would have to set up 191 00:18:22.680 --> 00:18:27.480 two observation spots that were far apart, or at least would seem to be. 192 00:18:30.319 --> 00:18:34.480 Reggio Montanus grasped that an astronomer does not have to travel in order to 193 00:18:34.519 --> 00:18:40.960 get two observation points that are in effect far apart. If the heavens rotate 194 00:18:41.000 --> 00:18:44.960 around the center of the universe, and if that center is at or near 195 00:18:45.000 --> 00:18:48.440 the center of the Earth, then the observation point of the astronomer who is 196 00:18:48.480 --> 00:18:53.599 on the surface of the Earth changes in its relationship to the heavens as they 197 00:18:53.640 --> 00:18:57.319 move. Simply because the astronomer is not looking at the heavens from the center 198 00:18:57.359 --> 00:19:00.359 of the universe, but from a point that is distant from the center. 199 00:19:02.880 --> 00:19:07.160 So imagine you're standing at the immediate dead center of a merry go round or 200 00:19:07.160 --> 00:19:15.160 a carousel on which horses are arranged in concentric circles. At the center is 201 00:19:15.160 --> 00:19:19.319 a stationary round platform, around which the circle of horses, each taking the 202 00:19:19.359 --> 00:19:26.000 same time to complete a circuit, revolve. As you look outwards and the 203 00:19:26.039 --> 00:19:30.519 horses turn around you, the relative position of the horses will remain the same. 204 00:19:30.599 --> 00:19:34.839 A horse which is in line with another horse at one moment will still 205 00:19:34.880 --> 00:19:41.000 be in line with it a quarter of a revolution later. But if you 206 00:19:41.119 --> 00:19:45.319 take a few steps in any direction until you reach the edge of the stationary 207 00:19:45.359 --> 00:19:52.359 platform, then the relative position of the horses will appear to change all the 208 00:19:52.440 --> 00:19:59.240 time perspective changes. Moreover, if you know the size of the stationary platform 209 00:20:00.039 --> 00:20:03.480 the distance to the outer ring of the horses, then you can use changes 210 00:20:03.519 --> 00:20:08.119 in relative position of the horses in the two other rings to work out how 211 00:20:08.160 --> 00:20:17.160 far away they are. In essence, Reggio Montanas figured out that you could 212 00:20:17.240 --> 00:20:22.880 measure the parallax of heavenly bodies by taking two observations from the same place but 213 00:20:23.000 --> 00:20:32.480 at different times, rather than taking two observations from different places but at the 214 00:20:32.519 --> 00:20:38.720 same time. Although Reggio Montanas worked out how to make such a measurement in 215 00:20:38.799 --> 00:20:45.960 fourteen seventy one, the full account of this procedure didn't get published until fifteen 216 00:20:45.000 --> 00:20:52.160 thirty one. Unfortunately, in fifteen forty eight, a text apparently by Reggio 217 00:20:52.240 --> 00:20:57.759 Montanas was published which claimed to measure the parallax of the comet which had appeared 218 00:20:59.000 --> 00:21:03.599 in fourteen seventy two, and to confirm that it was as close to the 219 00:21:03.640 --> 00:21:08.160 Earth because the paradox was a whopping six degrees, placing it much closer than 220 00:21:08.200 --> 00:21:15.559 the moon, which had a diurnal parallax about one degree. Now sadly, 221 00:21:15.400 --> 00:21:22.039 recent historians have established that this particular paper was not written by Reggio Montanas. 222 00:21:22.759 --> 00:21:27.319 It was found amongst his papers when he died. It was presumably in his 223 00:21:27.359 --> 00:21:33.720 handwriting. However, he didn't write it. He might have been working on 224 00:21:33.759 --> 00:21:37.920 it, he might have been copying something on it. But we'd know that. 225 00:21:37.960 --> 00:21:41.319 No one in the sixteenth century realized this, and that turned out to 226 00:21:41.319 --> 00:21:47.559 be the cause of a lot of confusion then in fifteen seventy two, the 227 00:21:47.599 --> 00:21:55.559 world changed again. Astronomer Tico Brahey noticed a new star in the sky. 228 00:21:55.880 --> 00:21:59.079 For a time, it was the brightest object in the heavens other than the 229 00:21:59.119 --> 00:22:03.519 Sun and the moon, brighter even than Venus. Such events only occur once 230 00:22:03.559 --> 00:22:08.000 in a thousand years or so, and unlike a comet, the new star 231 00:22:08.079 --> 00:22:15.640 stood still, which made it much easier to measure its parallax. All over 232 00:22:15.680 --> 00:22:21.880 Europe astronomers were literally obsessed with it. Since they now knew Reggio Montanus's real 233 00:22:21.960 --> 00:22:27.720 technique for measuring parallax, they naturally tried to apply it. Some found a 234 00:22:27.799 --> 00:22:36.880 measurable parallax, but others insisted that they couldn't it didn't exist accurately. Measuring 235 00:22:36.920 --> 00:22:41.440 parallax from far was easy, particularly as it required a more exact measurement of 236 00:22:41.480 --> 00:22:47.519 time than a sixteenth century clock could provide, but showing that there was no 237 00:22:47.599 --> 00:22:52.079 measurable parallax was much more straightforward. All one had to do was hold up 238 00:22:52.079 --> 00:22:56.319 a thread as a sighting device and find two stars that were exactly in line 239 00:22:56.359 --> 00:23:00.640 with this new one, but north or south of it. If the same 240 00:23:00.680 --> 00:23:04.880 stars were exactly in line with the nova later that same night, then there 241 00:23:06.039 --> 00:23:11.279 was no parallax. This simple technique was actually employed by the teacher of Johannes 242 00:23:11.400 --> 00:23:17.680 Kepler later on. And if there was no parallax, then the comet must 243 00:23:17.759 --> 00:23:22.279 be a vast distance away, far further than the Moon, whose parallax was 244 00:23:22.359 --> 00:23:27.759 quite easy to measure, and it must be super luminary, not a sublunary 245 00:23:27.799 --> 00:23:33.039 body. But how to explain the appearance of a new star in the heavens? 246 00:23:34.160 --> 00:23:38.559 Since there could be no natural explanation assuming the star was indeed the heavens, 247 00:23:40.200 --> 00:23:45.440 the event was just a miracle sent by God. The finest astronomers and 248 00:23:45.480 --> 00:23:51.640 astrologers, including Thomas Diggs in England, racked their brains in an attempt to 249 00:23:51.680 --> 00:23:56.599 figure out what it might portend, and hastened to publish their conflicting conclusions, 250 00:24:00.079 --> 00:24:04.000 making matters more complicated. The new star of fifteen seventy two was followed by 251 00:24:04.039 --> 00:24:10.599 a new Comment of fifteen seventy seven, and here again parallax measurement placed the 252 00:24:10.640 --> 00:24:15.079 comment far beyond the moon, where a nova or a new star could possibly 253 00:24:15.160 --> 00:24:21.319 regard it as a miracle. A comment was too commonplace to be handled in 254 00:24:21.359 --> 00:24:26.319 that way. Ra Hay worked out a further problem that could be solved by 255 00:24:26.400 --> 00:24:33.160 measuring parallax. A crucial difference between the Ptolemaic system on the one hand and 256 00:24:33.200 --> 00:24:37.799 the Copernican system on the other was that under these modern systems, Mars must 257 00:24:37.839 --> 00:24:45.400 at all times approach much closer to the Earth than under the Ptolemaic system. 258 00:24:45.680 --> 00:24:48.920 Ra Hay at first thought he had obtained a reliable figure for the parallax of 259 00:24:48.960 --> 00:24:56.519 Mars, which proved the Ptolemaic system was mistaken, although he later realized there 260 00:24:56.519 --> 00:25:03.559 were problems with that Reggio Montana. This procedure for measuring parallax ideally involved comparing 261 00:25:03.880 --> 00:25:11.200 the apparent position of a celestial object soon after dark with its apparent position not 262 00:25:11.319 --> 00:25:18.200 long before dawn, thus maximizing the parallax to be measured. Neither the New 263 00:25:18.279 --> 00:25:23.000 Star of fifteen seventy two nor the Comment of fifteen seventy seven set in the 264 00:25:23.119 --> 00:25:30.799 night sky viewed from northern Europe, so the ideal procedure was inapplicable. In 265 00:25:30.839 --> 00:25:34.039 the case of Mars. There was no choice but to make measurements when the 266 00:25:34.079 --> 00:25:38.359 planet was nearly in line with the Sun, and thus it never rose high 267 00:25:38.359 --> 00:25:45.079 above the horizon at night. In measuring the location of an object near the 268 00:25:45.119 --> 00:25:49.319 horizon, ray had to allow for the refraction caused by the greater thickness of 269 00:25:49.319 --> 00:25:55.759 the atmosphere which with its rays had passed, and eventually what he found was 270 00:25:55.799 --> 00:26:00.480 that he had miscalculated this. But as was the case in many times in 271 00:26:00.480 --> 00:26:08.480 the scientific Revolution, what mattered not was that the individual was wrong. The 272 00:26:08.519 --> 00:26:14.960 fact that tco Brian was wrong about Mars didn't make a difference. What mattered 273 00:26:15.079 --> 00:26:22.599 was his method was right, and the observations that he took day after day, 274 00:26:22.759 --> 00:26:29.119 night after night, proved instrumental for later astronomers who are able to correct 275 00:26:29.440 --> 00:26:34.960 the error. Again, this all gets down to whether or not Europeans are 276 00:26:36.000 --> 00:26:45.200 allowed to question knowledge under the ancient classical system. The answer is absolutely not. 277 00:26:45.200 --> 00:26:48.359 And so even if you don't understand any of this stuff about parallaxes and 278 00:26:48.400 --> 00:26:52.279 so on and so forth, and I only loosely grasp it, what you 279 00:26:52.400 --> 00:26:59.799 should come away with this is understanding that for the first time people were willing 280 00:26:59.839 --> 00:27:06.400 to take math and try to apply it in new, in unique ways to 281 00:27:06.480 --> 00:27:11.000 the world, in a way in which our understanding of the natural world became 282 00:27:11.319 --> 00:27:19.519 so much more systematized, so much easier to measure, and this is a 283 00:27:19.559 --> 00:27:26.160 great example of a few fundamental features of the scientific Revolution. The first is 284 00:27:26.279 --> 00:27:33.880 inevitability. Once Reggio Montanas developed a system for measuring parallax and published it, 285 00:27:34.680 --> 00:27:40.960 astronomers were set on a path that could only lead to proving Aristotle and Ptolemy 286 00:27:41.079 --> 00:27:48.200 were wrong. The second feature is time. Delays in publication meant that it 287 00:27:48.240 --> 00:27:55.359 took a long time for the inevitable conclusions to hit home, but ultimately that 288 00:27:55.400 --> 00:28:02.039 would not matter because of a shift in the mindset from Europeans from theorizing and 289 00:28:02.200 --> 00:28:10.039 logical argument to experimentation and objective proofs. Once a discovery had been made, 290 00:28:10.559 --> 00:28:17.559 there was no one making it, not that some didn't try. We'll get 291 00:28:17.599 --> 00:28:22.200 into the inquisition here in a few weeks when I discuss reactions to the scientific 292 00:28:22.240 --> 00:28:27.119 Revolution. And while I do not want to overly belabor the point, the 293 00:28:27.160 --> 00:28:33.720 second fundamental feature of the scientific revolution remains the impact of the printing press. 294 00:28:34.920 --> 00:28:42.119 Printing made it possible for Ray to survey a wide range of publications. Before 295 00:28:42.160 --> 00:28:48.119 he turned his gaze to the skies, he looked to a book, well 296 00:28:48.240 --> 00:28:52.839 multiple books, none of which would have been possible without printing. Historians of 297 00:28:52.880 --> 00:28:59.160 science have often and rightly suggested that the key to the scientific revolution is the 298 00:28:59.640 --> 00:29:06.000 math the midization of nature. Aristotle and Ptolema had assumed that the heavens were 299 00:29:06.000 --> 00:29:12.519 mathematically legible, and indeed Ptolemy had devised techniques for reading them. One aspect 300 00:29:12.519 --> 00:29:22.440 of the scientific revolution consists in the extension of mathematical theories to include sublunary phenomena. 301 00:29:22.880 --> 00:29:27.000 Where Aristotelian physics was preoccupied with qualities the four elements earth, air, 302 00:29:27.119 --> 00:29:33.279 fire, and water, and embody the four qualities hot, dry, cold, 303 00:29:33.319 --> 00:29:37.319 and wet. You'll remember that from the four humors, the new physics 304 00:29:37.680 --> 00:29:42.680 was preoccupied with movements and quantities that could be measured, and it quickly led 305 00:29:42.720 --> 00:29:48.559 to attempts to measure the speed of falling bodies, the speed of sound, 306 00:29:48.880 --> 00:29:53.400 and the weight of air. Where Aristotle had assumed that each element behaved differently, 307 00:29:55.200 --> 00:30:00.519 the new physics assumed that all heavy objects could be thought of as the 308 00:30:00.599 --> 00:30:07.720 same. Where Aristotelian physics had depended on all five senses, the new physics 309 00:30:07.240 --> 00:30:17.319 relied only on site. While traditional histories suggest that the mathematization of science started 310 00:30:17.480 --> 00:30:22.640 in the seventeenth century with new physics, really, we do get a glimpse 311 00:30:22.680 --> 00:30:30.759 of it much earlier through perspective painting. Galileo learned math from Hostilio Ricci, 312 00:30:30.960 --> 00:30:37.799 who by trade was a teacher of perspective to artists, not astronomers. The 313 00:30:37.920 --> 00:30:44.200 amazing thing is how an innovation in one field can now jump to another. 314 00:30:45.440 --> 00:30:52.640 It can lead to innovations in other, seemingly unconnected branches of science. In 315 00:30:52.720 --> 00:31:00.079 the sixteenth century, perspective and coordinators made the jump from perspective painting to geography. 316 00:31:00.079 --> 00:31:07.000 Now there wasn't necessarily anything new about this. Cicero had thought that geography 317 00:31:07.200 --> 00:31:14.839 was a branch of geometry. With geometry came abstraction. Perspective is represented with 318 00:31:14.960 --> 00:31:21.480 two parallel lines converging towards a vanishing point. It's in fact a grid used 319 00:31:21.480 --> 00:31:25.359 by artists to establish a picture plane, and the exact same thing can be 320 00:31:25.400 --> 00:31:32.240 done to establish latitude and longitude. Geometry, by the way, also acquired 321 00:31:32.640 --> 00:31:37.400 new importance as a result of the invention of gunpowder. Fortifications had to be 322 00:31:37.480 --> 00:31:42.240 built to resist cannonballs, which fly in straight lines at least as a bird 323 00:31:42.319 --> 00:31:48.480 season. In order to provide raking and flanking fire along every wall, forts 324 00:31:48.519 --> 00:31:55.240 needed to be designed on the page with very carefully measured angles and distances. 325 00:31:56.599 --> 00:32:02.359 Thus, if we were to ask how the scientific revolution became mathematized, the 326 00:32:02.440 --> 00:32:08.960 answer is clear. Perspective painting led to cartography, which led to navigation and 327 00:32:09.079 --> 00:32:15.240 proper astronomy, and ultimately to ballistics. And in many cases all of these 328 00:32:15.359 --> 00:32:23.240 changes in innovations happened simultaneously, with one technique jumping seamlessly from one branch of 329 00:32:23.279 --> 00:32:29.640 science to the next. These were like real things too, that made a 330 00:32:29.680 --> 00:32:34.720 difference in the actual world. In sixteen twenty two, for example, a 331 00:32:34.799 --> 00:32:39.640 fleet of Dutch ships tried to seize the Portuguese colony of Macao. A Jesuit 332 00:32:39.759 --> 00:32:46.519 mathematician did the geometry calculations to determine the distance to a stockpile of gunpowder that 333 00:32:46.640 --> 00:32:51.640 Dutch had brought ashore, and the angle of elevation at which the cannon should 334 00:32:51.640 --> 00:32:57.039 be set. A direct hit turned the tide of that battle and ensured that 335 00:32:57.160 --> 00:33:04.200 Macau remain a Portuguese colony. Bus If we ask how did the scientific revolution 336 00:33:04.279 --> 00:33:09.640 become mathematized, the answer is through the different fields that I've already laid out. 337 00:33:10.599 --> 00:33:14.359 And of course, and again I don't want to keep the laboring this 338 00:33:14.440 --> 00:33:20.440 point, but the answer is also because of printing, because printing allowed the 339 00:33:20.480 --> 00:33:25.000 spread of ideas from one branch of science to the next, and often within 340 00:33:25.039 --> 00:33:30.640 the same. Frankly, first and foremost, we might want to think about 341 00:33:30.680 --> 00:33:37.720 the scientific revolution as a revolt, a revolt by the mathematicians against the authority 342 00:33:37.759 --> 00:33:45.519 of philosophers, an authority they had held for thousands of years. We'll end 343 00:33:45.559 --> 00:33:51.240 it there for today, before continuing with the scientific Revolution next week. As 344 00:33:51.319 --> 00:33:54.640 always, if you would like additional content, click on any of the links 345 00:33:54.640 --> 00:34:00.119 in the show notes you can access the website. You can now do a 346 00:34:00.240 --> 00:34:07.640 free seven day trial of the Patreon accountants see what extra benefits patrons do acquire 347 00:34:07.720 --> 00:34:12.239 depending upon their tier, and if you'd like, you can check out a 348 00:34:12.280 --> 00:34:16.519 seven day free trial of Western Sieve two point zero. You can get the 349 00:34:16.599 --> 00:34:22.199 whole story go all the way back to Mesopotamia again from the beginning, but 350 00:34:22.360 --> 00:34:27.960 in much greater detail and with much better audio quality. At the outset