Scientific Method & Quantitative Reasoning

Rene Descartes – The RationalistDescribed the method to do science,known for his mind-body dualism

Major Works:• Discourse (1637) [full title: Discours

de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences]

• Meditations on first Philosophy (1641) [6 Meditations: Of the Things that we may doubt; Of

the Nature of the Human Mind; Of God: that He exists; Of Truth and Error; Of the Essence of Material Things; Of the Existence of Material Things; Of the Real Distinction between the Mind and the Body of Man]

Rene Descartes (1596-1650)

Rene Descartes’ Discourse

Describes the method to do science in a straightforward way (see below)

Major points:• Science must be based on

correct reasoning (logic)• Science must be formulated in

mathematical language

Starting line: “Good sense is the most evenly distributed thing in the world, for all people suppose themselves so well provided with it that even those who are the most difficult to satisfy

in every other respect never seem to desire more than they have.”

Modifying our Epicycle Model• More observation & reasoning shows that Mars does

not just go around an epicycle in 6 months• Also: loop generated by epicycle is not the same as

the size of the epicycle itself, see animation• Modify model to better describe Nature: Epicycle

turns once in a synodic period, i.e. time from one retrograde loop to the next

• See video

A Classical Example

• Aristotle observes that during lunar eclipses the Earth’s shadow on the moon is curved

• He assumes it will be curved for all eclipses• A hypothesis that explains this: the earth is round• A prediction of this theory is that the location of the

stars in the sky should be different for observers at different latitudes

• This is confirmed by additional observations– E.g. Canopus is visible in Egypt but not further north

• It is applied by Eratosthenes to determine the size of Earth

Scientific Literacy• An important skill to have in an increasingly

complex society• As a voting citizen you should know what you

are voting on • As a consumer you should know (or be able to

figure out) what you buy• Makes life more interesting• Nobody knows everything, but smart people can

figure out fast if something is obviously wrong

Quantitative Reasoning

• Amazingly powerful tool to understand the world around us

• Fundamentals:– Area &Volume– Scaling– Algebraic statements

Achieving Scientific Literacy(Arons Article)

• Two types of knowledge– Declarative (Learned Facts, “book knowledge”)– Operative (actually knowing how to solve problems)

• Trouble with GenEd courses– Too much in too little time– Getting a “feeling” for the subject doesn’t work– Need to understand the underpinnings first (area,

volume, scaling, energy, atoms,…)

How far away is the Moon?• The Greeks used a special configuration of

Earth, Moon and Sun (link) in a lunar eclipse• Can measure EF in units of Moon’s diameter,

then use geometry and same angular size of Earth and Moon to determine Earth-Moon distance

• See here for method

That means we can size it up!

• We can then take distance (384,000 km) and angular size (1/2 degree) to get the Moon’s size

• D = 0.5/360*2π*384,000km = 3,350 km

How far away is the Sun?

• This is much harder to measure!• The Greeks came up with a lower limit,

showing that the Sun is much further away than the Moon

• Consequence: it is much bigger than the Moon• We know from eclipses: if the Sun is X times

bigger, it must be X times farther away

Simple, ingenious idea – hard measurement

Another Example in Homework: The Cosmic Distance Ladder, Rung I• Now it is your turn to establish the length of the AU in kilometers.

First, calculate the distance Venus-Earth from a measurement of the travel time of a radar signal traveling from Earth to Venus and back in 4.72 minutes 42.5

•  million kilometers. Next, figure out the length of the AU by noting that the measurement was made while Venus and Earth were closest to one another. At this point, how close are Venus and Earth if we assume they are both at a distance from the Sun that is equal to the length of their respective semi-major orbital axes?  0.28AU 

Homework: The Cosmic Distance Ladder, Rung I

• Since we measured this distance in the first question in kilometers, you can now calculate the length of the Astronomical Unit in kilometers: 1 AU =   152 million km. 

• Since we made a couple of simplifying assumptions in our calculation, the answer is not very accurate. How far off is it?  2.01 million km