Many times we need to quickly illustrate the solar system or the planets. Usually we use photos to illustrate the planets. But sometimes the photos can be an over kill. Also to draw the entire solar system is a typical and can be used for illustrative purposes. Though most of the illustrations of the solar system in the school and other text books are horribly out of scale, (with no indication that the figure is not to scale!). For example, look at the illustration in the Class 6 Science Textbook from NCERT.
A good visualisation will always present a scale, and/or indicate whether the visualisation is to the scale or not. Though in the visualisation above the distances are given, they are not to scale.
Coming back to the topic of our post, a simple way to draw solar system diagram in latex is to use the PStricks package solarsystem. The package can create “Position of the visible planets, projected on the plane of the ecliptic” at a given time and date. This feature might be useful sometimes.
From the package manual:
As we can not represent all the planets in the real proportions, only Mercury, Venus, Earth and Mars are the proportions of the orbits and their relative sizes observed. Saturn and Jupiter are in the right direction, but obviously not at the right distance.
The orbits are shown in solid lines for the portion above the ecliptic and dashed for the portion located below.
The use of the command is very simple, just specify the date of observation with the following parameters, for example: \SolarSystem[Day=31,Month=12,Year=2020,Hour=23,Minute=59,Second=59]
By default, if no parameter is specified, \SolarSystem gives the configuration day 0 hours to compile.
The resulting output for the above code:
The output also provides the longitude and latitude of the planets at the time given.
Another package that is useful to create free standing planets is the tikz-planets package which we will see next.
One of the USPs of using LaTeX is the variety of packages that are available to get things done. Some packages will give you special environments to make your documents better, some will help in typesetting or some will help you create graphics or some just provide you with commands for specific symbols. Of course, all these can be done manually by creating your own command, but why reinvent the wheel? There are hundreds of packages at the Comprehensive TeX Archive Network. I have come across many packages that were useful via browsing the packages at CTAN. In this series of posts we will see some packages that are interesting and might be useful. This series of posts is also a sort of personal bookmarking scheme for me. It has happened in the past that I have discovered some interesting LaTeX package, only to forget its existence when I needed its functionality in a project.
In this first post, we will look at two related packages bohr and elements by Clemens Niederberger. The bohr package provides you with a simple functionality to draw the Bohr diagrams for different elements along with electronic configurations.
Load the bohr package by \usepackage{bohr} in the preamble
To use the package simple type the number of electrons and the element symbol. For example, Lithium \bohr{3}{Li} will simply give you
Similarly for other elements Lithium \bohr{3}{Li}Oxygen \bohr{8}{O}Carbon \bohr{12}{C}Mercury \bohr{80}{Hg}
Now another very useful option in the vohr package is to print the shell-wise electronic configuration for a given element. For example Oxygen \bohr{8}{O} \elconf{O} will give you
This will be a very useful feature when you are writing chemistry or atomic physics texts. Of course you can change the way the shells look. \setbohr{
shell-options-add = dashed, shell-options-add = red, shell-dist = .75em, nucleus-options-set = {draw=black,fill=orange,opacity=0.5}, electron-options-set = {color=green}, insert-missing} Mercury\bohr{80}{Hg} \elconf{Hg}
The insert-missing option will give you either the correct number of electrons when the element symbol is given, or will give you the element symbol when the number of electrons is given. There are more options to explore in the documentation.
Now let us look at the elements package.
This package provides means for retrieving properties of chemical elements like atomic number, element symbol, element name, electron distribution or isotope number.
The package provides atomic number, symbol, name, main isotope and electronic configuration for elements upto 118. For example, just using the atomic number 35 I can get \elementname{35} \elementsymbol{35} \elconf{35}
Having the data accessible in the form of number can be very useful especially if you want to generate tables. The table below from the package documentation was generated by iteratively looping atomic number and invoking commands
Such natural beauty! Does mathematics lie at the basis of these diverse and beautiful forms? Photo taken during summer of 2017 in Mumbai. None of these are native to India. On left: The Cannonball tree flower (Couroupita guianensis) is South and Central American, African Tulip Tree (Spathodea campanulata) native to tropical. Africa; On Right: The Cannonball tree flower, and Gulmohar is Madagascan (Delonix regia).
What could be more “organic” and “natural” than looking at a pristine forest with a variety of tree forms and leaf forms of various shapes and shades, with inflorescences of variety of shapes, sizes and colours? Mathematicians and physicists are often accused of being not able to enjoy nature and because mathematics and physical theory is so “abstract” and nature is so “organic”. Organic growth is in the form of variety of morphologies of roots, branches, flowers shapes and arrangement, leaf shapes and arrangements, while mathematics typically is abstract graphs, equations, symbols and numbers. How can these two possibly have anything in common? This has also to do with how biology is traditionally taught. While physics has mathematics at its foundation, the teaching of biology doesn’t acknowledge any need for mathematics – it is mostly descriptive as it was in its early stages a couple of centuries later. This is more so at the school level teaching of biology. So this creates an impression in the students and teachers alike that mathematics is not a part of “biological” nature and it is only reserved for falling bodies and ascending projectiles.
What can be similarities in the two images? One is abstracted representation of motion of a body in algebraic and graphical format and other is organic growth of a plant showing its branching and similar leaves with its pigmentation of chlorophyll.
Linnaean system brought order to seemingly diverse and chaotic forms of natural world. Linnaeus named the different forms. Naming is the first step in studying anything. Naming helps in categorisation, which is one of ways to formation of concepts. This led to further finer classification of the system as whole which now includes both flora and fauna. Then began the programme of finding organisms and classifying them in existing categories with descriptions – or creating new ones when the existing ones did not fit – became the normal way of doing biology in the nineteenth century. Even now finding a new plant or animal species is treated with celebrated as a new discovery.
Darwin in his thesis about evolution by natural selection used the differences and similarities of the form as one of evidence. He theorised that organisms that have evolved from common ancestors will show similar forms with slight variations. Over long periods of time these slight variations evolve into larger variations which ultimately leads to a completely different species. Fossil records tell us about ancestors and current relatives of organisms.
There is grandeur in this view of life, with its several powers, having been originally breathed by the Creator into a few forms or into one; and that, whilst this planet has gone circling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being evolved. (emphasis added)
The morphologies tell us about related species, the ancestries and divergences from there. The fossils tell us the ancestors, the missing links. So finding organisms, both extant and extinct, to fit in the jigsaw puzzle of tree of life became the standard programme in biology. This enabled us to construct the tree of life. Ernst Haeckel’s version of the tree, depicted below, is highly anthropocentric which places humans at the apex of evolution. This is rather common misconception about evolution – humans are not at apex of evolution or the prime product of it as some would have us believe – we have co-evolved with all the current extant species. Evolution by natural selection is not anthropocentric, it is indifferent to humans and other organisms alike. Daniel Dennett likens it to universal acid, and makes a point that it is not only applicable to living systems, but applies to any system which fulfil the three required criteria.
Haeckel’s – Pedigree of Man – a version of tree of life which is highly anthropocentric.
But can we make sense of similarities of the form in terms of mathematics? Can we find mathematical algorithms which will generate forms, as they generate trajectories of moving projectiles? Looking at similarities in form, it is Galileo who was one of the first to discuss the problem of scaling and its effect on form.
To illustrate briefly, I have sketched a bone whose natural length has been increased three times and whose thickness has been multiplied until, for a correspondingly large animals, it would perform the same function which the small bone performs for its small animal, From the figures here shown you can see how the proportion of the enlarged bone appears.
…
Whereas, if the size of a body be diminished, the strength of that body id not diminished in the same proportion; indeed the smaller the body the greater its relative strength. Thus a small dog could probably carry on his back two or three dogs of his own size; but I believe that a horse could not carry even one of his own size.
The kind of mathematical treatment that entered in study of biology by above classics looked at the mathematical aspects of morphological forms in organisms. The Curves of Life looks at the spiral forms which are found in nature, and also in various human creations – architecture and art.
The spiral is one of the most easily identifiable mathematical forms in nature.In many flowers, a double spiral forms the basis of the central pattern. The Fibonacci numbers are easily identifiable with this pattern.The spiral is found in animals too, most easily identifiable in the shells of various types. They represent logarithmic spirals.
Despite tremendous success of Darwin’s theory, physics and mathematics were in a separate compartment from biology. There seemed to be no common elements, while biology became more and more descriptive with focus on the form, but not mathematical.
The word “form” in this article will refer to the shapes of material objects, the arrangement in space of groups of them, and the arrangement in space of their component parts. Our appreciation of form is partly sensory, but we can be helped by measurement and calculation to gain some confidence that what we perceive is not entirely unconnected with the outside world. (Physical Principles underlying Inorganic Form – S.P.F. Humphreys-Owen)
But this is a folly. Nature and organic growth is as mathematical as is the description of a projectile flying under gravity. Perhaps the mathematical description is
The sparse branching of Frangipani (Plumeria sp.), a native of central America.The dense branching of the Acacia (Vachellia nilotica) native to Africa, Middle East and India. Is the branching in Frangipani and Acacia related? What about the branching in grass in the figure at top? Can these be generated from a single mathematical algorithm? And why do only these forms are found and not any other?
In my experience a lot of young children who take to biology do so because they hate mathematics or computations. In India there are even streams at +2 level which allow you to shun mathematics for biological subjects. This utter hatred for mathematics is, IMHO, due to a carelessly designed and too abstracted mathematics curriculum at the school level – a curriculum which takes out the soul of mathematics and puts on a garish display of the cadaver of mathematics with bells and whistles. But this post is not about the problems of mathematics education, I have talked about it elsewhere.
The aim of this series of posts is to touch upon the inherent mathematics and algorithms in the natural world. How nature is mathematical especially in living and non-living things. How can algorithms generate natural forms? In the next posts in this series we will explore how the ideas of mathematical models can explain the variety of forms that result from natural selection in environment and possibly why only those forms can be found.
Note: All photographs were taken by me over the years. Only now I am able to piece a narrative linking them together.
. . . the most ubiquitous finding in developmental research is that infants show more adult- like performance as they grow older. [1]
The very fact that children grow up and become adults relates to the above sentence. The starting and the ending points of the child’s development are known to us. The main aim of the developmental theories is to find out the ‘paths’ that lead to the change from an infant to an adult. Thus in a way different theories ‘map’ out the regions between the infant and adult. For achieving this, every theory has some tools, processes, structures and concepts. Change and development in each of these parameters results in the overall development of the child. The parameters and the agents of development may be different in the different approaches. We consider each of the major developmental theories with respect to their parameters of development.
The broad outlines for the various developmental approaches presented here follow closely the section What Develops? at the end of each chapter in [4] unless otherwise indicated.
1 Piagetian Approach
The basic paradigm that the Piagetian approach envisages, is the stagewise development of the child and the associated psychological structures or schemes. The stages of child range from an infant at sensorimotor stage to an adolescent in formal operational stage. Associated with each stage is the characteristic structural change in schemes, regulations, functions, and various logico-mathematical structures. So the answer to the question ‘what develops’ according to Piaget would be that the schemes and structures associated with each stage develop, in accordance to characteristic for each stage. This development can be assessed through observations, interviews taken by the experimenter [4] pg. 72.
2 Information Processing Approach
In the information processing approach, the cognitive processing is the measure of development. The increase in cognitive processing means that it becomes efficient, well organized, and the content of information also increases, which results in the overall development. Children acquire ‘rules’, ‘strategies’, ‘scripts’ and more knowledge. The concept of memory is directly related to the cognitive processing, it determines the ‘speed’ of processing as well as the ‘output’. So the increase in the memory capacity results in the overall increase ‘quality’ as well as the ‘quantity’ of the cognitive processing. In case of the connectionist approach the strengthening of connections in terms of number and strengths over time, would represent the development of the particular path of connections related to the input.
3 Vygotskian Approach
In the Vygotskian approach the development of the child has a distinctly social character. Also the development is not just limited to the individual, but is much broader in the outlook; viz. a culture, a species, a child, a cognitive skill. The basic unit of development is the “active-child-in- cultural-context.” This unit is responsible for construction of different cognitive skills, including “system of meaning and its psychological tools.” The ideal end point in development of each culture is dependent of the goals of the particular culture. The goal of the culture is the basic driving force for the development of the child, and the interactions of the child with the society are responsible for this. The psychological tools or the higher mental functions are the parameters of the development of the child. A volitional control, conscious awareness of these higher mental functions represents a final step in the process of development [6] Chapters 5 and 6.
4 Psychoanalytic Approach
In the Freudian or the psycho-analytic approach three structures viz. the id, ego and the superego form the central basis of the theory. The id is the largest portion of the mind, is innate and is responsible for biological needs and desires. The id aims to satisfy the impulses without any delay. The ego which emerges in early infancy, is the conscious part of the personality and is responsible for the completion of id’s impulses in accordance with reality. The superego develops between 3 -6 years and incorporates the values of the society. The emergence, interaction and the struggle between these three structures form the basis of development. [2] pg. 14, [4] pg. 137.
5 Social Learning Theories
The learning theorists provide only a few universal behaviors as the act of learning itself depends on ‘what the environment has to offer.’ Since this theory accounts for development primarily as a quantitative change, one in which the learning episodes accumulate over time; the ability to skillfully learn what is observed or listened from the other people or by attending to symbolic characters or imitation in the society is developed in the children universally [4] pg. 201.
6 Ecological Theories
In the Gibson’s ecological theory child actively learns from experience and environment. The child learns to detect the structure, which specifies the information available to be perceived. Gibson has proposed four parameters for human behavior viz. agency, prospectivity, search for order, and flexibility. Agency “is the self in control, the quality of intentionality in behavior.” We see ourselves as distinct from the environment, and can be agent to cause the change in it. Thus with development our aspect towards this relationship changes. Prospectivity refers to the intentionality, planning and anticipation of the future. This is also seen to develop with the age. The search for order would involve the search for patterns, order and regularity in trying to make the sense of the environment. The aspect of flexibility comes into picture with the adaptation to the environment with whatever ‘skills’ one has. The affordances [“what an environment offers it provides for an organism; they are opportunities for action”] needed for working in another setting are obtained by changing the activities [4] pg. 360.
7 Modularity Nativism
The term modularity nativism refers to a set of approaches that postulate certain innate modules, structures or constraints, each specialized for a particular domain of cognition [3] pg 20. The modules are ‘pre-programmed’ to respond to specific sorts of information. These innate modules require a ‘trigger’ in form of little experiences, with appropriate content, to be activated. The different modules are posited to be relatively independent of each other, such that the development in one does not overflow into another. The developmental changes in thinking are caused by external factors such as maturation [4] pg. 427. This in turn implies that the infant mind is not very different from that of an adult.
8 Theory Theory
The theory theory approach is another domain specific approach to child development, which likens the children’s knowledge to a scientific theory [3] pg. 20. The children are capable of constructing intuitive, folk , everyday na ̈ıve “theories” for a particular domain [4] pg. 423. According to this theory the child has different theories for different domains. In the development process the children ‘test’ these intuitive theories, just like a scientists, in light of their experiences, thus they are like ‘little scientists’. So the answer to the question, What Develops? is that these intuitive na ̈ıve theories develop, with the experience of the children with the real world.
9 Dynamic Systems
The dynamic systems approach to child development addresses change over time in the complex holistic systems, especially self organizing ones [4] pg. 432. The term dynamic system most generally means “simply systems of elements that change over time.” In dynamic systems we have two basic themes for development [5] pg. 563:
Development can only be understood as the the multiple, mutual, and continuous interaction of all the levels of the developing system, from the molecular to the cultural.
Development can only be understood as nested processes that unfold over many time scales, from milliseconds to years.
One of the metaphors that is used to explain the dynamic systems approach is a mountain stream . The behavioral pattern are analogous to the eddies and the ripples of a mountain stream. In mountain stream metaphor “behavioral development is seen as an epigenetic process, that is truly constructed by its own history and system wide activity” [5] pg. 569. Thus development is seen as a process in which new behavioral patterns emerge because of interaction. [5]
References
[1] Aslin as quoted in [3] pg. 47.
[2] Berk L., Child Development 3rd Ed. 2001, Prentice Hall of India
[3] Flavell J. H., Miller P. H., Miller S. A. Cognitive Development 4th Ed. 2001, Prentice Hall
[4] Miller P. H., Theories of Developmental Psychology 2001, W.H. Freeman
[5] Thelen E., Smith L. B., “Dynamic Systems Theories” Chapter 10 in Handbook of Child Psychology : Vol. 1. Theoretical Models of Human Development 1998, Wiley
[6] Vygotsky L. S., Thinking and Speech Ed. Rieber, Carton The Collected Works of L.S. Vygotsky, Vol. 1: Problems of General Psychology 1987 Plenum
We generally associate cooking with its product – the edible food it produce. But here I would like to reflect on the process of cooking. For me, personally, cooking is a stress relieving exercise. The final product is one of the components not the only one. What makes great food great is not only the product, but the process – how well the cook has balanced the flavours, the textures and aromas. And this is something that the cook savours. For a good cook the best satisfaction comes from the orgasmic faces that eating the food brings out….
So let us look at what are various steps in the process.
Deciding what to cook?
This can be a major challenge sometimes, especially when you have abundant ingredients at your disposal. But if you have limited ingredients the kind of food that you can cook is obviously limited. You cannot make a chicken-noodle soup if you don’t have chicken and noodles, rest of the ingredients are optional in way. chicken noodle soup
Gathering the ingredients
This I think is one of the feel good parts for me. Getting all the ingredients ready, washing and cleaning them. This also creates a sense of what will be the output. One can visualise the metamorphosis of the raw elements to something well cooked – how their colour, flavour and texture will change during the cooking. When you look at cooking as a process, a lot of imagination has to be therein you as a cook
The Spices and flavours
The natural flavours in the food are sometimes too bland or we need to temper them to suit our palate. Different cuisines uses different spices and herbs to make the food relishable. Lot of powders or different spices are used, while in some cases whole spices are used. The ginger garlic paste is another basic ingredient in many cuisines. And finally salt. Almost no dishes are made without salt. Put too little or none it becomes bland or tasteless, put a bit too much and it becomes salty!
The plate below has onion cut very finely, coriander cut, ginger garlic thick paste, black pepper powder, salt, paprika, chilly powder, coriander powder, and cumin powder. spice set
The garam masala is another quintessential routine for much of indian cooking.You can make one yourself easily. garam masala
Pepper and chillies of various kinds are used to add fire to the food. Choose yours wisely. chillies
Fresh or dried herbs are a also used to give the required zest. Basil is the best! basil
The Cutting Board
I prefer wooden cutting board. It is easy on the knives and it feels good to cut on it.
Mincing an onion in little pieces with a knofe which obeys and performs with movement of your hands is one of best experiences. onion and garlic cut
Though onion can be cut in several ways, depending on what you are trying to make it taste like… red onion cut
A baby cabbage is one of the most wonderful things to cut. To see the internal structure of the cabbage and you know it is going to be a nice dish.. cabbage cut
Halving mushrooms or slicing them is therapeutic in itself. You feel victorious after halving a packet of them. You feel something has been achieved for the day. mushroom cut
cut carrot onion and sun dried tomatoes
The cutting tools – the knives – are a separate topic in themselves and deserve their own separate post.. But as they say a good chef will not trade the knives with (think of something which rhymes with knives)…
Setting the stage
Once the ingredients are ready, cleaned and cut, our next step begins. This is the preparation for. the final act of cooking itself. It may be mixing things, marinating them or mashing them up. preparation of chicken marination
marinating chicken with curd and spices
putting the chicken on the spikes and put on fire test
and see the marinated raw chicken metamorphise to chicken malai kabab
The Cooking Medium
Each dish has to be cooked in the media it is meant to be cooked with. Depending on the cuisine it can be groundnut oil, coconut oil, mustard oil, butter, olive oil, sesame oil, ghee or lard. Each medium will give its unique flavour and aroma to the food. And some dishes don’t taste the same or even taste awful when not cooked in the preferred media.
When you add the masala to the medium, the way it releases its colours and aromas is something that I admire. The aroma of onions getting cooked in pure ghee… yum.. ir garlic getting cooked in olive oil…
The Cooking Vessel
Over last couple of years I have been introduced to cast iron vessels for cooking and I am not going back. I have already planned not to purchase any non-stick ones in the future. It is cast iron all the way now. cast iron dutch oven cooking onion
chicken cooking in a cast iron dutch oven on a braai
The Order of Things
When the medium is hot enough you start to add things. In the order, sometimes you change the ordery you get good result, or it is a fiasco. I think new dishes were discovered in this way. When you go outside the cookbook and try something new, many times it might be a failure, but at times you will be rewarded with great new dishes.
For example, you would want to make the flour ready for making roties or bread before you want to make them, flour for making breads vegetable roast in the oven – do we add herbs and spices first or later? vegetable roast in the over – do we add herbs and spices first or later?
The Aromas
You know you have a great dish incoming when the dish starts to release its aroma. You might now smell it if you are too close to a dish while cooking, go to a different room and come abck you will know its a good dish cooking. A good aroma also creates a sense of mystery and longing for the dish, it stirs up appetites. aroma from a chicken curry
Unfortunately our science of smell is not that well developed so I cannot make you experience the aromas – but use your imagination..
The Colours and texture
The colours in the food make it attractive. A dull coloured food will be not that attractive. That is reason we find salads so much attractive! colourful salad – with texture and taste colourful salad – with texture and taste colourful salad – with texture and taste and hummus tomato onion uttappa looks refreshing so does two egg omelette!
The Presentation
As we approach the final stages of cooking the presentation of the food is the final frontier. No matter how well you have cooked it, if the food is not presented well, well it loses it’s charm, though it might be still edible.. biryani with raita and salad assorted sauteed vegetables hummus, pita bread, couscous salad, pickles couscous vegetable salad ghee rice with dry fruits and caramelized onions pork pandi curry aka coorg pork with naan and curd salad (raita) boiled eggs with paprika and fresh coriander
The Taste
The real satisfaction to a cook comes from the pleasure that other find in eating the food that has been cooked. It pays of all the effort that was put in process, while the product is something akin to the metaphorical tip of the iceberg..
It is always joyful to cook for others than cooking alone for yourself. It helps you relax as you order the seemingly raw ingredients into a masterful recipe full of flavour, aroma and texture. Rightfully done cooking is the most therapeutic exercise for your troubled soul. It is perhaps this reason that mothers while cooking the food want you to eat more and more…
We had earlier posted illustrations from Raja Shiv Chatrapati from the book Raja Shiv Chatrapati.
I have managed to create a cleaner pdf book of these images.
You can get them here.
[archiveorg raja-shiv-chatrapati width=560 height=384 frameborder=0 webkitallowfullscreen=true mozallowfullscreen=true]
Following is an excerpt from Beyond the Planet Earth by the Russian scientist and science fiction author Konstantin Tsiolkovsky about the life in 2017, from approximately 100 years ago.
State of Humanity in 2017
What was our Earth like in the year 2017, in which our story is set?
On all Earth there was one beginning: congress, consisting of elective representatives from all states. It had existed already more than 70 years and resolved all questions concerning humanity. Wars were impossible. Misunderstandings between people were settled by peaceful means. Armies were very limited. Actually, these were armies of labor. Population, with happy enough conditions in the last one hundred years, was trebled. Trade, technology, art, and agriculture attained significant success. Huge metallic dirigibles, lifting thousand of tons, communicated, transported goods conveniently and inexpensively.
Especially beneficial were the huge air ships, sending inexpensive loads, such as trees, coal, metals, and so forth downstream with the wind almost for nothing. Aircraft served for especially fast transfer of small number of passengers or precious cargo; the most commonly used were airplanes for one or two persons.
Humanity marched peaceful on the path of progress. However, fast growth of population forced all thinking people and rulers concern.
Ideas regarding the possibility of technical conquest of use of world deserts were considered long ago, — still more than one hundred years ago. In 1903, one Russian thinker wrote a serious work on this matter and proved mathematically, on the basis of the scientific data of that time, the complete possibility of settlement of solar system. But these ideas were almost forgotten, and only our company of scientists revived them and partly carried them out.
Let us first talk about the technological changes he has predicted. He predicts that airships or dirigibles would be a major form of transport for cargo, while aircrafts are used for smaller cargo and passengers. Also, there is an indication in the last paragraph that some space travel has happened.
The political and social scene is something that is off the charts. Tsiolkovsky shows a lot of optimism in how humans are able to resolve their differences and bring forth peace and happiness in all. I think this is a reflection of the zeitgeist of the era. Lot of revolutionary changes were happening which resulted in progress in the fields in almost all fields. So it wouldn’t be a deviation to think that humans would also progress along the same lines and resolve their differences peacefully to live a prosperous and happy life.
You can read Beyond the Planet Earth here as a part the his collected works.
John Tukey‘s wisdom on importance and value of graphics and pictures in making sense of exploring data.
Consistent with this view, we believe, is a clear demand that pictures based on exploration of data should force their messages upon us. Pictures that emphasize what we already know — “security blankets” to reassure us — are frequently not worth the space they take. Pictures that have to be gone over with a reading glass to see the main point are wasteful of time and inadequate of effect. The greatest value of a picture is when it forces us to notice what we never expected to see. (p. vi emphasis in original)
A short account of Galileo’s description of his own experiment on accelerated motion — a short account of it, the apparatus he used and the results he got.
The first argument that Salviati proves is that in accelerated motion the change in velocity is in proportion to the time (𝑣 ∝ 𝑡) since the motion began, and not in proportion to the distance covered (𝑣 ∝ 𝑠) as is believed by Sargedo.
“But for one and the same body to fall eight feet and four feet in the same time is possible only in the case of instantaneous (discontinuous) motion; but observation shows us that the motion of a falling body occupies time, and less of it in covering a distance of four feet than of eight feet; therefore it is not true that its velocity increases in proportion to the space. (Salviati)
Also, he proves that the increase in proportion is not of simple doubling but larger. They agree upon a definition of uniformly accelerated motion,
“A motion is said to be equally or uniformly accelerated when, starting from rest, its momentum receives equal increments in equal times. (Sargedo)
To this definition Salviati adds an assumption about inclined planes, this assumption is that for a given body, the increase in speed while moving down the planes of difference inclinations is equal to the height of the plane. This also includes the case if the body is dropped vertically down, it will still gain the same speed at end of the fall as it would gain from rolling on the incline This assumption makes the final speed independent on the profile of the incline. For example, in the figure below, the body falling along𝐶 → 𝐵, 𝐶 → 𝐷 and 𝐶 → 𝐴 will attain the same final speed.
This result is also proved via a thought experiment (though it might be feasible to do this experiment) for a pendulum. The pendulum rises to the height it was released from and not more.
After stating this theorem, Galileo then suggests the experimental verification of the theorem. of The actual apparatus that Galileo uses is an wooden inclined slope of following dimensions: length 12 cubits (≈ 5.5 m, 1 cubit ≈ 45.7 cm), width half-cubit and three-finger breadths thick . In this plank of wood, he creates a very smooth groove which is about a finger thick. (What was the thickness of Galileo’s fingers?) The incline of this plank are changed by lifting one end. A bronze ball is rolled in this groove and time taken for descent is noted.
“We repeated this experiment more than once in order to measure the time with an accuracy such that the deviation between two observations never exceeded one- tenth of a pulse-beat.
Then Galileo performed variations in the experiment by letting the ball go different lengths (not full) of the incline and “found that the spaces traversed were to each other as the squares of the times, and this was true for all inclinations of the plane”. Each variation was repeated hundreds of times so as to rule out any errors. Also, the fact that for different inclines the times of descent were in noted and were in agreement with the predictions.
Since there were no second resolution clocks to measure time, Galileo devised a method to measure time using water. This was not new, water clocks were used earlier also.
The basic idea was to the measure the amount of water that was collected from the start of the motion to its end. The water thus collected was weighed on a good balance.This weight of water was used as a measure of the time. A sort of calibration without actually measuring the quantity itself: “the differences and ratios of these weights gave us the differences and ratios of the times”
Galileo used a long incline, so that he could measure the time of descent with device he had. If a shorted incline was used, it would have been difficult to measure the shorter interval of time with the resolution he had. Measuring the free fall directly was next to impossible with the technology he had. Thus the extrapolation to the free fall was made continuing the pattern that was observed for the “diluted” gravity.
“You present these recondite matters with too much evidence and ease; this great facility makes them less appreciated than they would be had they been presented in a more abstruse manner. For, in my opinion, people esteem more lightly that knowledge which they acquire with so little labor than that acquired through long and obscure discussion. (Sargedo)
