The Tumultuous Astronomical Revolution

Abstract

Although the astronomy of Ptolemy laid a solid foundation for future scientific research, Europe would see a period of severe scientific decline over a millennium following the astronomer’s death.

朝聞道, 夕死可矣

Hear the Way in the morning,

and it won’t matter if you die that evening,

-----------Book 4 (“Humaneness”), Analects, by Confucius (551–479 BC).

Translated by Burton Watson (1925–2017)

Appendices

Appendix 3.1: Indirect Evidence of the Earth’s Rotation

We live on a spherical planet that rotates on its axis once every 24 h, but we have no idea that the ground is actually moving. Our bodies have been physiologically adapted to this dynamic environment, and therefore the fact that Earth is rotating is hidden from us. Since the time of Copernicus, scientists have proposed new theories to describe Earth’s motion. Now we know that Earth revolves around the Sun and rotates on its axis, and that Earth’s equatorial plane is inclined to the ecliptic plane at an angle of about 23.5^°. Scientists have been exploring ways to prove, right here on the surface of Earth, that the planet is indeed rotating. Below we present two natural phenomena and an experimental device. By analyzing their principles of operation or interpreting their mechanisms, we will be able to verify Earth’s rotational motion.

• (1) Tropical Cyclones: Counterclockwise Rotation in the Northern Hemisphere

Suppose a typhoon of moderate intensity, located at 30^° N on the Pacific Ocean, has a radius of 300 km and an eye of radius 30 km. The typhoon’s maximum wind speed is typically about 80 m/s, which occurs at the eyewall, as shown in the following diagram. Below we will derive a wind speed distribution model and then apply it to estimate the maximum wind speed around the eye.

Suppose a typhoon has formed at 30^° N. Figure 3.5 shows a magnified diagram of a part of the typhoon. We can create plane polar coordinate system with the center of the cyclone’s eye as the origin and use it to describe the motion of any fluid particle within the storm, with its speed expressed as

Fig. 3.5

Dynamic structure of a tropical cyclone on Earth’s surface

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {v} = v_{r} \hat{\iota }_{r} + v_{\theta } \hat{\iota }_{\theta } = \dot{r}\hat{\iota }_{r} + r\omega \hat{\iota }_{\theta }$$

(3.16)

Here $\dot{r} = dr/dt$ and $\omega = d\theta /dt$. Now, let’s assume that Earth rotates at an angular velocity of $\Omega$, which equals to 15^°/h or $15 \times 2\uppi/(360 \times 3600) = 7.27 \times 10^{ - 5} /{\text{s}}$. Since the typhoon’s point mass is located at 30^° N, the angular velocity of Earth’s rotation at this point has the following orthogonal component: $\Omega \sin 30^{ \circ } \,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{\text{i}}_{\text{z}} = 0.5\,\Omega \,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{\text{i}}_{\text{z}}$. At this time, the point mass is subject to the effect of the Coriolis force and has a magnitude of

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {f} = - 2m(0.5\Omega )\hat{\iota }_{z} \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {v} = m\Omega \left( {r\omega \hat{\iota }_{r} - \dot{r}\hat{\iota }_{\theta } } \right)$$

(3.17)

We substitute the data given above into Eq. (3.17), which expresses a relationship in mechanics, as follows. The change of angular momentum of the point mass $d(mr^{2} \omega )/dt$ should then be equal to the torque created by the Coriolis force exerting on the point mass in its the tangential direction $\vec{r} \times \vec{f} = - {\text{m}}\Omega {\text{r}}\frac{dr}{dt}\hat{\iota }_{z}$, thus

$$d\left( {mr^{2} \omega } \right) = - m\Omega rdr$$

(3.18)

Assuming that the typhoon’s flow field is rotationally symmetric, the velocity of the fluid particles at a distance of r from the typhoon center can be regarded as the wind speed at that location. Therefore, if we integrate the expression (3.18) from $r = r_{2}$ (the typhoon’s outermost radius) to $r = r_{1}$ (the boundary of the typhoon’s eye, or the eyewall’s radius), then we obtain the following relationship:

$$\upomega_{1} = \frac{\Omega }{2}\left( { \frac{{{\text{r}}_{2}^{2} }}{{{\text{r}}_{1}^{2} }} - 1} \right)$$

(3.19)

In the process of arriving at the integral (3.19), we have assumed that the angular velocity of the eye’s point mass is $\upomega_{1}$, and the angular velocity of fluid particles in the periphery of the typhoon $\upomega_{2} = 0$. We now substitute ${\text{r}}_{1} = 30\,{\text{km}}$ and ${\text{r}}_{2} = 300\,{\text{km}}$ into the formula to obtain $\upomega_{1} = 3.6 \times 10^{ - 3} /{\text{s}}$. Since this is a positive quantity, it indicates that the rotation is counterclockwise.

Next we will estimate the wind speed around the eyewall. The tangential velocity of a fluid particle on the eyewall is ${\text{r}}_{1} \times\upomega_{1} = 30 \times 10^{3} \times 3.6 \times 10^{ - 3} = 108\,{\text{m}}/{\text{s}}$. This quantity is about 20% greater than the observed (typical) value of 80 m/s mentioned earlier, which suggests that the Coriolis force that drives the typhoon constitute a rather significant factor. Other factors that affects the typhoon’s movement include the friction between the storm and the sea’s surface, evaporation and rise of water vapor at the eye, precipitation within the typhoon’s chamber, and the obstruction of the typhoon’s periphery, all of which having contributed roughly 20% to the observed speed.

• (2) Atmospheric Circulation

In the absence of Earth’s rotational motion, the air closer to the equator would rise after being heated by the Sun. As this heated air reaches a certain altitude, it would drift northward and southward to the polar regions until it cools down, at which time the air would drift back to the equatorial region, forming a north-south atmospheric circulatory system. Due to Earth’s rotation, however, the atmospheric structure is a much more complex situation. At the upper reaches of the Pacific Ocean and the Atlantic Ocean, we can designate an atmospheric circulatory zone bounded by the 30^° and 60^° latitudes (either in the Northern or Southern Hemisphere). We will then employ a simple point- mass motion model to estimate the wind speed of atmospheric circulation at 30^° latitude in order to confirm the importance of the Coriolis force generated by Earth’s rotational motion on atmospheric circulation.

Again let’s refer to Fig. 3.5. Suppose a point mass m, located at latitude $\theta$ and longitude $\varphi$ in the upper atmosphere, is in motion along a direction parallel to Earth’s surface. The motion generated by the Coriolis force under Earth’s rotation satisfies the equations

$$m\frac{{dv_{x} }}{dt} = (2m\Omega \sin \theta )v_{y}$$

(3.20)

$$m\frac{{dv_{y} }}{dt} = - (2m\Omega \sin \theta )v_{x}$$

(3.21)

Here the velocity along the latitude is $v_{x} = R\cos \theta \frac{d\varphi }{dt}$ and the velocity along the longitude is $v_{y} = R\frac{d\theta }{dt}$. Substituting these two velocity expressions into (3.20) and (3.21), we obtain

$$\frac{d}{dt}\left( {\cos \theta \frac{d\varphi }{dt}} \right) = 2\Omega \sin \theta \frac{d\theta }{dt}$$

(3.22)

$$\frac{d}{dt}\left( {\frac{d\theta }{dt}} \right) = - 2\Omega \sin \theta \cos \theta \frac{d\varphi }{dt}$$

(3.23)

Rearranging (3.22), we have

$$\frac{d}{dt}\left( {\cos \theta \frac{d\varphi }{dt} + 2\Omega \cos \theta } \right) = 0$$

(3.24)

Integrating (3.24) with respect to time, we obtain

$$\cos \theta \frac{d\varphi }{dt} + 2\Omega \cos \theta = C_{1}$$

(3.25)

Here $C_{1}$ represents the constant of integration. Substituting (3.25) back into (3.28), we have

$$\frac{d}{dt}\left( {\frac{d\theta }{dt}} \right) = - 2\Omega \sin \theta \left( {C_{1} - 2\Omega \cos \theta } \right)$$

(3.26)

Now we multiply both sides by $d\theta$ to get

$$\frac{d\theta }{dt}d\left( {\frac{d\theta }{dt}} \right) = 2\Omega \left( {C_{1} - 2\Omega \cos \theta } \right)d(\cos \theta )$$

(3.27)

Integrating (3.27) with respect to time, we have the expression

$$\frac{1}{2}\left( {\frac{d\theta }{dt}} \right)^{2} + C_{2} = 2\Omega \cos \theta \left( {C_{1} -\Omega \cos \theta } \right)$$

(3.28)

Here $C_{2}$ is another constant of integration. When $\frac{{{\text{d}}\uptheta}}{\text{dt}} = 0$, the direction of the wind returns to the latitude’s east-west orientation. In other words, the wind no longer blows toward the north or south pole. The latitude $\uptheta^{ *}$ at this time represents the highest latitude the wind is able to reach, and the following applies:

$$C_{2} = 2\Omega \cos \theta^{*} \left( {C_{1} -\Omega \cos \theta^{*} } \right)$$

(3.29)

Now, we will need boundary conditions in order to compute the two constants of integration. First of all, if we assume that the wind along the equator only blows toward the north or south pole (i.e., along the longitude’s north-south orientation), then its velocity v satisfies the relationship $\left( {\frac{d\theta }{dt}} \right)_{0} = \frac{v}{R}$. On the other hand, the velocity of the wind blowing in the easterly or westerly direction (i.e., along the latitude) should be 0, or $\left( {\frac{d\varphi }{dt}} \right)_{0} = 0$. Substituting this condition into (3.25), we get $C_{1} = 2\Omega$, where $\Omega$ is the angular velocity of Earth’s rotational motion. Substituting this constant into (3.28), we arrive at the following

$$- \left( {\frac{d\theta }{dt}} \right)_{0}^{2} = 4\Omega ^{2} \left( {2\cos \theta^{*} - \cos^{2} \theta^{*} - 1} \right) = - \left[ {2\Omega \left( {1 - \cos \theta^{*} } \right)} \right]^{2} ,$$

(3.30)

from which we obtain

$$\theta^{*} = \cos^{ - 1} \left[ {1 - \frac{v}{{2R\Omega }}} \right]$$

(3.31)

Since we know that 30^° N represents a boundary of the atmospheric circulatory zone, i.e., $\theta^{*} = 30^{ \circ }$, if we substitute this expression into the above, we can conclude that the wind speed at this particular latitude should be $v = 448\,\text{km}/\text{h}$.

According to what we know today, the atmosphere at this latitude contains a jet stream (alternatively, subtropical jet). The jet stream’s speed is generally between 200 and 300 km/h, and can increase to 400 km/h occasionally. Our calculations above result in a value larger than the usual estimate. The reason is that actual atmospheric circulation involves a robust, three-dimensional vortex, which is subject to depletion from the friction of viscous fluids and the phase transition between liquid water and water vapor. These complexities are not modeled by our smooth, two-dimensional configuration with non-viscous fluids. However, the simplified model discussed here emphasizes that, although inaccurate, the estimated value computed from the Coriolis force generated by Earth’s rotation is nevertheless sufficiently close to the actual value measured. Such Coriolis force has a considerable effect on atmospheric circulation. To conclude, the calculation of atmospheric circulation also provides indirect evidence of Earth’s rotational motion.

• (3) Foucault Pendulum

The principles behind the Foucault pendulum, which we will employ to confirm Earth’s rotation, can be explained using one of the following two approaches: the method of mechanics⁴⁴ and the geometric approach.⁴⁵ Now we will explore the method of mechanics as follows.

Consider a simple pendulum placed at latitude $\varphi$. Assuming that the pendulum swings at a very small amplitude, the motion of the bob’s point mass can be regarded as that on a level surface parallel to the latitude. An xy (two-dimensional) rectangular coordinate can therefore be superimposed on the location with the origin being the bob when the swing amplitude is zero. Assuming also that the length of the pendulum is l, then the resultant force (force of gravity plus the tension of the pendulum’s string or rod) can be decomposed on the xy Cartesian coordinates as $- mg \times \frac{x}{l}$ and $- mg \times \frac{y}{l}$, respectively. Now, the vertical component of the angular velocity Ω of Earth’s rotation on the Cartesian plane is $\omega =\Omega \sin \varphi$. Substituting this into the Coriolis force formula, we obtain the effective Coriolis force component

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{F} = - 2m\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{\Omega } \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {v} = 2m\omega \left( {\dot{y}\widehat{{\varvec{\iota}_{\varvec{x}} }} - \dot{x}\widehat{{\varvec{\iota}_{\varvec{y}} }}} \right)$$

(3.32)

In the above formula, $\widehat{{\varvec{\iota}_{\varvec{x}} }}$ and $\widehat{{\varvec{\iota}_{\varvec{y}} }}$ are the unit vectors in the x and y directions, respectively. The motion equations are therefore

$$m\ddot{x} = - mg \times \frac{x}{l} + 2m\omega \dot{y}$$

(3.33)

$$m\ddot{y} = - mg \times \frac{y}{l} - 2m\omega \dot{x}$$

(3.34)

Adding (3.33) to (3.34) multiplied by i, we can take advantage of the complex polar coordinates $(r,\theta )$, $z \equiv x + iy = re^{i\theta }$, and combine the two expressions

$$\ddot{z} + 2i\omega \dot{z} + \frac{g}{l}z = 0$$

(3.35)

The general solution to (3.35) can be written as

$$z = e^{ - i\omega t} \left( {Ae^{{i\sqrt {\omega^{2} + g/l} t}} + Be^{{ - i\sqrt {\omega^{2} + g/l} t}} } \right)$$

(3.36)

From (3.36), we can see that when the Coriolis force is absent, the solution (3.36) becomes

$$z = Ae^{{i\sqrt {g/l} t}} + Be^{{ - i\sqrt {g/l} t}} ,$$

(3.37)

which is the general solution to the ordinary single pendulum problem, and the angular frequency is $\sqrt {g/l}$. In the case where the Coriolis force is present, the angular frequency becomes $\sqrt {\omega^{2} + g/l}$, and $e^{ - i\omega t}$ as a multiplicative factor is added to the front. This implies that in addition to swinging, the plane on which the pendulum swings will also rotate gradually in a counterclockwise direction, and this angular speed of the rotation will be $\omega =\Omega \sin \varphi$, where $\Omega = 360^{ \circ } /{\text{day}}$ is the angular speed of Earth’s rotation. Substituting Ω into the expression we obtain

$$\upomega = \frac{{360^{ \circ } \sin {\upvarphi }}}{\text{day}}$$

(3.38)

If the location is the equator, ${\upvarphi } = 0,$ and the plane of the single pendulum’s motion will not rotate. At the north or south pole, ${\upvarphi } =\uppi/2,$ and the plane of the single pendulum’s motion will rotate once per day. The reason for its rotation is the Coriolis force generated by Earth’s rotation. Again the fact that Earth rotates is indirectly confirmed.

Apart from solving the differential equation directly, Somerville [30] applied a brilliant geometric method to directly observe the angular velocity of the plane of the pendulum’s motion, see please the Fig. 3.6. As explained by Somerville, one may consider the case where the single pendulum is located at point C on the latitude $\varphi$. Suppose Earth has rotated an angle of $\delta \alpha$ after a period of time. During this time, the position of the pendulum has moved from ${\text{C}}_{2}$ to ${\text{C}}_{1}$. Consider the angle $\angle {\text{C}}_{1} {\text{AC}}_{2}$. If this angle is viewed from the plane of the vertical pendulum’s swinging motion, namely the plane where the triangle $\Delta {\text{C}}_{1} {\text{BC}}_{2}$ lies, then the swinging direction of the pendulum on this plane should move in parallel. The swinging direction of the pendulum at ${\text{C}}_{1}$ will therefore still be parallel to ${\text{C}}_{2} {\text{B}}$, i.e., the angle of δβ will form between ${\text{C}}_{2} {\text{B}}$ and ${\text{C}}_{1} {\text{B}}$. From the geometric relationship shown in Fig. 3.6, we will therefore obtain

Fig. 3.6

Geometric analysis of Foucault pendulum’s motion. Based on the information of Fig. 3.5 of Somerville [30]

$$\overline{{C_{1} C_{2} }} = \overline{{AC_{1} }} \times \delta \alpha = \overline{{BC_{1} }} \times \delta \beta$$

(3.39)

Since $\frac{{\overline{{AC_{1} }} }}{{\overline{{BC_{1} }} }} = \sin {\upvarphi },$ we have $\delta \beta = \delta \alpha \sin \varphi$. After integrating this expression, a rotation of $\alpha = 360^{ \circ }$ per day by Earth implies a rotation of $\beta = 360^{ \circ } \sin \varphi$ per day by the Foucault pendulum.

Appendix 3.2: Principles behind Galilei’s Telescope Designs

The Galilean telescope employs a convex lens with a focal length of ${\text{f}}_{1}$ as the object lens and a concave lens with a focal length of ${\text{f}}_{2}$ as the eyepiece. The principle behind the formation of an image with this configuration is illustrated and explained in the diagram below. Assuming that the effective radius of the object lens is ${\text{r}}_{1}$ and the effective radius of the eyepiece is ${\text{r}}_{2}$, the following relationship applies

$$\frac{{{\text{f}}_{1} }}{{{\text{f}}_{2} }} = \frac{{{\text{r}}_{1} }}{{{\text{r}}_{2} }} = {\text{M}}$$

(3.40)

Here M is also the magnification of the telescope. Since an image often appears blurry around the lens’ edges, they are excluded from the calculation of the effective radius. Consequently, the effective radius of a lens will be smaller than its actual physical radius (Fig. 3.7).

Fig. 3.7

A Galilean telescope’s structure

Now we consider the first telescope constructed by Galilei to spot the ships far away from the Venetian coast. It had a magnification factor of 9×, or ${\text{M}} = 9$. Suppose the radius of the pupil of a human eye is 3 mm, and the effective radius ${\text{r}}_{2}$ of the eyepiece is twice that of the pupil, or 6 mm. Substituting these values into the above formula, we obtain ${\text{r}}_{1} = 9 \times 6 = 54\,{\text{mm}}$. Assuming that the actual radii of the object lens and the eyepiece are twice their respective effective radii, then this telescope is equipped with an eyepiece of actual radius 12 mm and an object lens of actual radius 108 mm. The telescope’s actual length, in this case, has nothing to do with the focal length. Now suppose that the telescope’s body is not excessively long for the purposes of transportation and installation. Then we can assume that its focal length is ${\text{f}}_{1} = 1\,{\text{m}}$, in which case ${\text{f}}_{2} = (1/9)\,{\text{m}}$. Here ${\text{f}}_{1}$ is the minimum length of the telescope’s main body (the tube). If shading parts in front of the object lens and behind the eyepiece are taken into consideration, the maximum length would perhaps be 1.1 m. Galilei reportedly used spherical cannonballs to grind and polish his concave lens, so ${\text{f}}_{2} = (1/9)\,{\text{m}}$ could have been the cannonball’s radius, which is reasonable considering that the inner diameter of the artillery barrel in Galilei’s time was comparable in size. Taking into account of these factors, the design specifications of the telescope described above appear to be reasonable.

Now we consider the telescope Galilei employed to observe Jupiter and the Moon. It had a magnification factor of 20×. Again we adopt the same specifications for the eyepiece, as in the previous telescope, with ${\text{r}}_{2} = 6\,{\text{mm}}$ and ${\text{f}}_{2} = (1/9)\,{\text{m}}$. The corresponding object lens would have the specifications ${\text{r}}_{1} = 20 \times 6 = 120\,{\text{mm}}$ and ${\text{f}}_{1} = 20/9\,{\text{m}}$. The length of this telescope would therefore be over 2 m, with the diameter of the object lens approaching 25 cm. This would have been a rather large and bulky telescope, and with its size and weight it would have been best regarded as a telescope for an indoor observatory rather than a mobile or transportable instrument. A large-diameter objective lens would also be better suited for astronomical observations, as it provides a field of view as wide as 30 arc minutes, which is required to observe the Moon.

As Galilei might have used cannonballs to grind and polish his concave lenses, we can speculate that the quality of lenses could not have been too high. The scattering of light caused by the lenses’ rough surface and the chromatic aberration caused by white light passing through the lens would have produced blurred images. In addition, as shown in the photograph of a Galilean telescope below, the diameter of the eyepiece is perhaps close to 123 mm, as mentioned earlier, but the diameter of the object lens is obviously much smaller than 120 mm. It is therefore reasonable to believe that the images seen by Galilei were probably quite blurry and the field of view could not have been too wide. What Galilei saw were probably only portions of the objects he tried to observe and not their entirety.

Appendix 3.3: The Work of Scientific Revolutionaries

Continental Europe in the 16th century witnessed the fervent cultural, artistic, political and economic growth of the Renaissance and the Scientific Revolution, which brought about rapid improvement in the quality of spiritual and material civilization in the history of humankind. Traditional manufacturing methods were no longer adequate to meet the needs of society, and a variety of new inventions would soon become reality. One important invention was vacuum-producing machinery, a practical invention providing an effective substitute for human and animal labor. Vacuum machines were used for pumping and extracting groundwater and for driving simple farming machinery. Even military equipment for besieging cities found use for them. The originators of vacuum engineering were from Italy, the birthplace of the Renaissance.

All well-drilling operators in Florence in the 16th century were aware that the suction pumps they had been using to extract groundwater could reach no deeper than 32 feet (or 18 cubits) underground. Evangelista Torricelli (1608–1647), one of Galilei’s last students, conjectured that this limit was the result of the maximum thrust the weight of the air could exert on the well water. To prove his theory, Torricelli designed an experiment in which a vertical glass tube filled with mercury had its upper end sealed off and its lower end exposed to air. Due to atmospheric pressure, the mercury column remained in the tube, but the maximum height of the mercury column reached only 30 in. (760 mm). Any mercury above this height would flow out of the tube via its lower end. Dividing the weight of the mercury column by the cross-sectional area of the glass tube gives what is now called one atmosphere (a unit of pressure).

Subsequently, Frenchman Blaise Pascal (1623–1662) extended Torricelli’s experiment and reasoned that if the mercury tube was placed at a location of a higher altitude, such as the top of a mountain, then the height of the mercury column would be reduced, as it was common knowledge that air is thinner at higher altitudes (the lighter the air, the lower the pressure). Pascal conducted a series of experiments between 1648 and 1651 to prove his conjecture. At about the same time, Robert Boyle (1627–1691), the son of Anglo-Irish nobleman Richard Boyle, 1st Earl of Cork, was obsessed with science, even though he had put himself in extreme danger by choosing to side with parliamentarian forces in the 1640 s when the English Civil War broke out. Though perhaps a bit eccentric with his fascination with science, Boyle was a lifelong, devout Christian, so whenever he discovered a new phenomenon he would attribute it to the Creator. Boyle continued the vacuum experiments of Torricelli and Pascal, using an improved vacuum pump designed by his student Robert Hook (1635–1703) to extract the air entirely from the glass tube. He discovered that the height of the mercury column in the pressure gauge connected to the container was reduced. From the relative changes between the height reduction and the container’s volume, he proposed what would be later referred to as Boyle’s law: the pressure of a gas in a container multiplied by its volume is a constant.

Apart from practical principles concerning vacuum, geometry was also one of the greatest facilitators of the Scientific Revolution, having existed for over two millennia and had a much wider scope of practical applications. One of the most famous geometers was the philosopher René Descartes (1596–1650). Descartes was born into a French aristocratic family and studied law at the University of Poitiers. He joined the Dutch States Army of Maurice, Prince of Orange, during the Dutch War of Independence (Eighty Years’ War) against the rule of Habsburg Spain. After the war, Descartes settled in the Dutch Republic in 1628 and devoted himself to scientific research. In 1649, Descartes was invited by the Queen of Sweden to Stockholm to be her tutor. He contracted pneumonia early in the following year due to the cold weather and died days later. Interestingly, Descartes only spent the last 20 years of his entire 53 years of life engaging in scientific research, but he managed to make notable contribution in a number of disciplines, including meteorology, mechanics, astronomy, mathematics and optics. As a result, he was regarded as one of the key figures in the Scientific Revolution.

The greatest contribution of Descartes is the creation of analytical geometry, a mathematical method that can be applied to describe geometric curves or surfaces and to compute the point(s) of intersection of two curves or the curve(s) of intersection of two surfaces. Descartes’ achievements opened the door to using equations to describe relationships between quantities, as well as computational processes. The days of using purely verbal descriptions for these purposes were numbered, and the classical Euclidean approach for drawing geometric diagrams was no longer the only method available to make geometric arguments. In physics, Descartes described the relationship between the angles of incidence and refraction when light passes through a boundary between two different isotropic media. He further used this relationship to elaborate on the formation of rainbows. This was a time when the fact that white light is in fact a combination of lights of different (roughly seven) wavelengths in the visible spectrum was not widely known. Nevertheless, Descartes was able to apply the relationship between the angles of incidence and refraction to calculate the angular differences between the seven colors. The fact that Descartes was able to come up with values quite close to what are currently known is remarkable indeed. Overall, Descartes was lauded by the philosophical community for his approach to scientific research, but opinions regarding his practice of science were less favorable. As an example, France’s delayed acceptance of Newton’s universal law of gravitation for decades can be attributed to the many French scientists’ adherence to Descartes’ physical theories that did not agree with Newton’s.

At last, there is another natural philosopher who remained an influential figure during the Scientific Revolution, so we are including him on the list of scientific revolutionaries: Englishman Francis Bacon (1561–1626). Bacon was the son of Sir Nicholas Bacon (1510–1579), Lord Keeper of the Great Seal of England. After graduating from Trinity College in Cambridge, Francis Bacon became quite successful in law, politics, and diplomacy, as his father had been. He was created Baron Verulam in 1618 and was promoted to the office of Lord High Chancellor at the same time and later Viscount St. Alban. In 1626, Bacon died of pneumonia, reportedly after having contracted the condition while studying the effects of freezing on meat preservation. Bacon’s philosophy in science focused on empirical research; some would even characterize this emphasis as a kind of extreme empiricism, in stark contrast to Plato’s philosophical approach. He not only disapproved of Aristotelianism but also rejected both the Ptolemaic and Copernican cosmological views, since these doctrines offered no pragmatic value to him. Instead he believed that scientists ought to collect all the useful knowledge and phenomena hidden in nature as comprehensively as possible, so that humans would be able to recover their God-given abilities to control nature which had been lost since Adam’s expulsion from the Garden of Eden. In the 17th and 18th centuries, Bacon’s scientific accomplishments were compared to those of Plato and Aristotle. In retrospect, however, no scientific achievements can be unequivocally attributed to Bacon, let alone his impact on future generations of scientists.