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A few months ago, I embarked on a job search. In addition to the usual litmus test, considering the product, team, role, etc., I was adding some complexity. I wanted to continue my work as an educator. Last year, my research interests had taken me as far as Chile, where I taught computer science in public high schools as a Fulbright Scholar. My research question was simple — what should skills-based education look like? Back in San Francisco, I was disappointed to find there weren’t all that many EdTech companies that inspired me. Enter . On the surface, the product might not look like it has much to do with education, but I saw an opportunity. Parabola gives knowledge workers the capabilities of engineers—a visual way of creating scripts to analyze data. Parabola This happens all the time. You have data living in a few different places—lead data in Salesforce, performance data in Dropbox, and a smattering of other attributes living in Google sheets. You want to perform some analysis on them, and then move the results somewhere else. “More likely than not, you’ll want to perform the same analysis at the end of every month. Usually, you are faced with two options. You could get really fast at doing the same things over and over again. Maybe allow yourself to dabble in the guilty pleasures side of your Spotify to make it through the final stretch. Your only other good option is bothering an engineer and convincing them this is worth it. Parabola offers a bridge between manual work and fully engineered solutions. Instead of repeating processes or allocating g resources, Parabola offers a remedy where ‘mood-boosting’ Spotify playlists do not. engineerin How to build flows with Parabola By creating a ‘flow,’ a series of operations you can perform on your data, you can automate away a lot of the work you’re stuck repeating. Parabola’s proposed paradigm is: build a data pipeline once, so that you can run it whenever you want, thereby cutting out the tedium of repeated work I think of Parabola as an educational tool, one that makes engineering design patterns accessible Although Parabola users are not writing code, they are to think like engineers. Creating a flow in Parabola requires many of the patterns that allow engineers to build robust systems. learning design Creating data pipelines in Parabola is a study in how to design, build, and test systems. I like working on a product that deepens a user’s knowledge base, instead of shying away from the challenge of putting powerful tools in the hands of users. But more importantly, Parabola serves as a sandbox, where users learn these design patterns through experimentation Over the course of my research project in Chile, I became enamored with the , which postulates that people learn most effectively through experimentation and play. In my workshops, students built their first video games using ‘sandbox-style’ educational tools developed by MIT. Constructionist theory of learning Although it’s tempting to break things into bite-sized chunks for users or students, we’re often over simplifying complex concepts. I found that teachers consistently underestimate their students’ ability to figure things out on their own. In my programming workshops, I saw the power of experimentation as a teaching method. I saw what students could build when equipped with the right tools. Back to Parabola. I liked the idea of giving people the power to engineer solutions. But I loved the tactic: trust people to figure it out for themselves. And that’s exactly what I saw Parabola users doing. It’s inspiring to see the complexity of flows that previously non-technical users build for themselves in Parabola. The team has put a ton of thought into building a product you can tinker with. The interface was designed to make powerful operations easy to understand conceptually. Whereas Excel was never designed for the crazy things it’s used for today, Parabola was. As a result, Parabola users are programming concepts intuitively. Parabola users are creating fully-engineered solutions all on their own. As an educator, that got (and continues to get) me really excited. learning

Why I joined Parabola

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Scientific American Supplement, No. 365, December 30, 1882 by Various, is part of the HackerNoon Books Series. You can jump to any chapter in this book . RECENT HYDRAULIC EXPERIMENTS. here RECENT HYDRAULIC EXPERIMENTS. At a late meeting of the Institution of Civil Engineers, the paper read was on "Recent Hydraulic Experiments," by Major Allan Cunningham, R.E. This paper was mainly a general account of some extensive experiments on the flow of water in the Ganges Canal, lasting over four years—1874-79. Their principal object was to find a good mode of discharge measurements for large canals, and to test existing formulæ. There are about 50,000 velocity, and 600 surface-slope measurements, besides many special experiments. The Ganges Canal, from its great size, from the variety of its branches abounding in long straight reaches, and from the power of control over the water in it, was eminently suited for such experiments. An important feature was the great range of conditions, and, therefore, also of results obtained. Thus the chief work was done at thirteen sites in brickwork and in earth, some being rectangular and others trapezoidal, and varying from 193 ft. to 13 ft. in breadth, and from 11 ft. to 7 in. in depth, with surface-slopes from 480 to 24 per million, velocities from 7.7 ft. to 0.6 ft. per second, and discharges from 7,364 to 114 cubic feet per second. For all systematic velocity measurements, floats were exclusively used, viz., surface floats, double floats, and loaded rods. Their advantages and disadvantages had been fully discussed in the detailed treatise "Roorkee Hydraulic Experiments"—1881. They measured only "forward velocity," the practically useful part of the actual velocity. The motion of water, even when tranquil to the eye, was found to be technically "unsteady;" it was inferred that there is no definite velocity at any point, and that the velocity varies everywhere largely, both in direction and in magnitude. The average of, say, fifty forward velocity measurements at any one point was pretty constant, so that there must be probably average steady motion. Hence average forward velocity measurements would be the only ones of much practical use. To obtain these would be tedious and costly, and special arrangements would be required to obviate the effects of a change in the state of water, which often occurred in a long experiment, as when velocities at many points were wanted. As to surface-slope its measurement—from nearly 600 trials—was found to be such a delicate operation that the result would be of doubtful utility. This would affect the application of all formulas into which it entered. The water surface was ascertained, on the average of its oscillations, to be sensibly level across, not convex, as supposed by some writers. There were 565 sets of vertical velocity measurements combined into forty-six series. The forty-six average curves were all very flat and convex down stream—except near an irregular bank—and were approximately parabolas with horizontal axes; the data determined the parameters only very roughly; the maximum velocity line was usually below the service, and sank in a rectangular channel, from the center outward down to about mid-depth near the banks. Its depression seemed not to depend on the depth, slope, velocity, or wind; probably the air itself, being a continuous source of surface retardation, would permanently depress the maximum velocity, while wind failed to effect this, owing to its short duration. On any vertical the mid-depth velocity was greater than the mean, and the bed velocity was the least. The details showed that the mid-depth velocity was nearly as variable from instant to instant as any other, instead of being nearly constant, as suggested by the Mississippi experimenters. The measurement of the mean velocity past a vertical was thought to be of fundamental importance. Loaded rods seemed by far the best for both accuracy and convenience in depths under 15 ft. They should be immersed only 0.94 of the full depth. The chief objection to their use, that—from not dipping into the slack water near the bed—they moved too quickly, was thus for the first time removed. A double float with two similar sub-floats at depths of 0.211 and 0.789 of the full depth would also give this mean with more accuracy and convenience than any instrument of its class; this instrument is new. Measurement of the velocity at five eighths depth would also afford a fair approximation. One hundred and fourteen average transverse velocity curves were prepared from 714 separate curves. These average curves were all very flat, and were convex down stream—over a level or concave bed—and nearly symmetric in a symmetric section. The velocity was greatest near the center, or deepest channel, decreased very slowly at first toward both banks, more rapidly with approach to the banks or with shallowing of the depth, very rapidly close to the banks, and was very small at the edges, possibly zero. The figure of the curve was found to be determined by the figure of the bed, a convexity in the bed producing a concavity in the curve and , and more markedly in shallow than in deep water. Curves on the same transversal, at the same site, and with similar conditions, but differing in general velocity, were nearly parallel projections. At the edges there was a strong transverse surface flow from the edge toward mid-channel, decreasing rapidly with distance from the edge. The discussion showed that it was almost hopeless to seek the geometric figure of the curves from mere experiment. vice versa Five hundred and eighty-one cubic discharges were measured under very varied conditions. The process adopted contained three steps: (1) Sounding along about fifteen float courses, scattered across the site in eight cross sections; time, say four hours. (2) Measurement of the mean velocities through the full depths in those float courses, each thrice repeated; time, say four hours. (3) Computation, say two hours. This process was direct and wholly experimental; each step was done in a time which gave some chance of a constant state of water. From an extended comparison of all results under similar conditions, it appeared that the above process yielded, under favorable circumstances, results not likely to differ more than 5 per cent. The sequel showed that in a channel with variable regimen, a discharge table for a given site must be of at least double entry, as dependent on the local gauge-reading, and on the velocity or surface-slope. Special attention was paid to rapid approximations to mean sectional velocity. The mean velocity past the central vertical, the central surface velocity, and Chézy's quasi-velocity—i.e., 100 × √( R × S ) where R=the hydraulic mean depth, and S=surface slope—were tried in detail; thus 100, 76, and 83 average values thereof respectively were taken from 581, 313, and 363 detail values. The ratios of these three velocities to the mean velocity were taken out, and compared in detail with Bazin's and Cutter's coefficients. Other formulæ were contrasted also in slight detail. Kutter's alone seemed to be of general applicability; when the surface slope measurement is good, and the rugosity coefficient known for the site—both doubtful matters—it would probably give results within 7½ per cent. of error. Improvement in formulæ could at present be obtained only by increased complexity, and the tentative research would be excessively laborious. Now the first two ratios varied far less than the third; thus their use would probably involve less error than the third, or approximation would be more likely from direct velocity measurement than from any use of surface slope. The connection between velocities was probably a closer one than between velocity and slope; the former being perhaps only a geometric, and the latter a physical one. The mean velocity past the central vertical was recommended for use, as not being affected by wind; the reduction coefficient could at present only be found by special experiment for each site. Three current meters were tried for some time with a special lift, contrived to grip the meter firmly parallel to the current axis, so as to register only forward velocity, and with a nearly rigid gearing wire. No useful general results were obtained. Ninety specimens of silt were collected, but no connection could be traced between silt and velocity; it seemed that the silt at any point varied greatly from instant to instant, and that the quantity depended not on the mean velocity, but probably on the silt in the supply water. Forty measurements of the evaporation from the canal surface were made in a floating pan, during twenty five months. The average daily evaporation was only about 1/10 in. The smallness of this result seemed to be due to the coldness of the water—only 63 deg. in May, with 165 deg. in the sun and 105 deg. in shade. Lastly, it must suffice to say that great care was taken to insure accuracy in both fieldwork and computation. About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books. This book is part of the public domain. Various (2006). Scientific American Supplement, No. 365, December 30, 1882. Urbana, Illinois: Project Gutenberg. Retrieved https://www.gutenberg.org/cache/epub/18763/pg18763-images.html This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at , located at . www.gutenberg.org https://www.gutenberg.org/policy/license.html

RECENT HYDRAULIC EXPERIMENTS

On the Economy of Machinery and Manufactures by Charles Babbage is part of the HackerNoon Books Series. You can jump to any chapter in this book . Of the Identity of the Work When It is of the Same Kind, and its Accuracy when of Different Kinds here Chapter 10. Of the Identity of the Work When It is of the Same Kind, and its Accuracy when of Different Kinds 79. Nothing is more remarkable, and yet less unexpected, than the perfect identity of things manufactured by the same tool. If the top of a circular box is to be made to fit over the lower part, it may be done in the lathe by gradually advancing the tool of the sliding-rest; the proper degree of tightness between the box and its lid being found by trial. After this adjustment, if a thousand boxes are made, no additional care is required; the tool is always carried up to the stop, and each box will be equally adapted to every lid. The same identity pervades all the arts of printing; the impressions from the same block, or the same copperplate, have a similarity which no labour could produce by hand. The minutest traces are transferred to all the impressions, and no omission can arise from the inattention or unskilfulness of the operator. The steel punch, with which the cardwadding for a fowling-piece is cut, if it once perform its office with accuracy, constantly reproduces the same exact circle. 80. The accuracy with which machinery executes its work is, perhaps, one of its most important advantages: it may, however, be contended, that a considerable portion of this advantage may be resolved into saving of time; for it generally happens, that any improvement in tools increases the quantity of work done in a given time. Without tools, that is, by the mere efforts of the human hand, there are, undoubtedly, multitudes of things which it would be impossible to make. Add to the human hand the rudest cutting instrument, and its powers are enlarged: the fabrication of many things then becomes easy, and that of others possible with great labour. Add the saw to the knife or the hatchet, and other works become possible, and a new course of difficult operations is brought into view, whilst many of the former are rendered easy. This observation is applicable even to the most perfect tools or machines. It would be possible for a very skilful workman, with files and polishing substances, to form a cylinder out of a piece of steel; but the time which this would require would be so considerable, and the number of failures would probably be so great, that for all practical purposes such a mode of producing a steel cylinder might be said to be impossible. The same process by the aid of the lathe and the sliding-rest is the everyday employment of hundreds of workmen. 81. Of all the operations of mechanical art, that of turning is the most perfect. If two surfaces are worked against each other, whatever may have been their figure at the commencement, there exists a tendency in them both to become portions of spheres. Either of them may become convex, and the other concave, with various degrees of curvature. A plane surface is the line of separation between convexity and concavity, and is most difficult to hit; it is more easy to make a good circle than to produce a straight line. A similar difficulty takes place in figuring specula for telescopes; the parabola is the surface which separates the hyperbolic from the elliptic figure, and is the most difficult to form. If a spindle, not cylindrical at its end, be pressed into a hole not circular, and kept constantly turning, there is a tendency in these two bodies so situated to become conical, or to have circular sections. If a triangular-pointed piece of iron be worked round in a circular hole the edges will gradually wear, and it will become conical. These facts, if they do not explain, at least illustrate the principles on which the excellence of work formed in the lathe depends. About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books. This book is part of the public domain. Charles Babbage (2003). On the Economy of Machinery and Manufactures. Urbana, Illinois: Project Gutenberg. Retrieved https://www.gutenberg.org/cache/epub/4238/pg4238.html This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at , located at . www.gutenberg.org https://www.gutenberg.org/policy/license.html

Of the Identity of the Work When It is of the Same Kind, and its Accuracy when of Different Kinds

Welcome everyone with another problem to solve! Today we have some nice math to work with: factorials! We are going to build an algorithm to : we’ll see how fast they grow and how big they get, and finally, we will calculate the sum of their digits to solve our problem. calculate huge factorials Today’s problems are given by : a super cool website to look at if you are in the mood for some head-scratching! There are more than 800 math and coding problems to solve and a vast archive of discussions about them. It is literally the perfect tool to train your math skills and learn something new. Go check it out and try to solve your challenges too! ProjectEuler.net Enough talking: let’s see what today’s problem has to offer. n! means n * (n-1) * … * 3 * 2 * 1. For example, 10! = 10 * 9 * … * 3 * 2 * 1 = 3628800, and the sum of the digits of 10! is 3+6+2+8+8+0+0 = 27. Find the sum of the digits in the number 100! As we anticipated before, what we are asked to do is to , in particular, the number 100! = 100 * 99 * … * 3 * 2 * 1 : pretty easy...right? Well, not quite! calculate the digits of a number Let’s put down some basics then and see how to manage such a big product. Factorials! The particular notation stated by the problem is a famous concept in math known as . The factorial of a number is simply . It’s usually expressed with the expression mark after the number (n!) or, in symbols, as factorial the product of all the natural numbers up until that number So, for instance, the factorial of the number 10, written as 10!, is 10! = 10 * 9 * … * 3 * 2 * 1 = 3628800. What’s really cool about factorials is since we are used to dealing with Big-O complexity notation, here’s a chart comparing some function’s growth rates. It’s pretty clear that the factorial growth is the : for an , the number of operations executed by a factorial growing function exceeds (by a lot) a thousand operations! how fast they grow: worst instance size of less than 10 One simple way to see this behavior is to : if you still have your scientific calculator from high school, like the famous Casio or Sharpe, you can easily see that : it gives you back a math error. This is because the number cannot be stored by the memory of the calculator, for it gets . try with a calculator you can’t calculate over 69! too big to address Some more recent calculators try to go over that bond: for example, the calculator of my phone (a Redmi 7) can calculate up to 170!, giving out the result with the exponential notation (7.25741562e³⁰⁶) if you try with 171! though, it just spits a disappointing ∞ … which is really not the case. It’s also really cool to look at how many digits the result of a factorial has: the number 100! has 158 digits the number 500! has 1135 digits the number 1000! has 2568 digits the number 5000! has 16326 digits The number 10000! has 35660 digits: it is so big that I could not fit it in a single image. Here’s a GIF with all the digits: But wait a minute: didn’t we just say that we could not calculate over the factorial of 70? There’s a trick to do so that we’ll see in a moment. But to understand how we can calculate arbitrarily big factorials, we first must go back to the basics of multiplication. How we multiply When you were kids, chances are that your teacher at school taught you to multiply in columns with a process similar to this: On the left side, a simple multiplication between a three-digit number and a one-digit number, in which each digit from the right to the left gets multiplied by 9. At each multiplication, if the result is greater than 9, the remainder is written on top of the next digit (right to left) and added on the next step of the process; on the right side, a multiplication of a three-digit number and a two-digit number, where there are two “temporary” results in between of the process. Here, we previously wrote a 0 as the rightmost digit in the second temporary result and summed those to get the final result. It’s really basic math, but this process works because we can the starting number in units, tens, hundreds, and so on, and multiply each of those by the second number. After that, we just sum each individual partial result to obtain the final result. Something like this: decompose That is precisely what we are going to do on a much bigger scale. We are going to deal with very big numbers, so this process needs to be modified a bit to work with the integer’s memory limits built into the language we are using (Python, in this case). To be true, Python 2 used to have a size limit in it’s type, but since Python 3 replaced it, the limit was somehow removed. Other languages do have it, though, so we build this approach to keep the algorithm portable to those languages too. Also, it will come in handy to multiply from the right to the left. int Coding the process Basically, we first convert our number to a string and reverse it. After that, we multiply each digit for the second number, add the previous remainder, and store the result and the new remainder for the next calculation. Let’s see the process step by step first: With this process, ideally, the only two limits we have on our products are: : when the number we are trying to multiply gets too big, we’ll be thrown a , for the interpreter has not enough memory to allocate the calculated number. Every modern computer has 8, 16, or 32Gb of RAM tough, so you should have plenty of memory to allocate before reaching the limit. The RAM limit MemoryError as we’ll see better later, the time needed to complete the process grows pretty quickly. Even if it’s not a real limit (one can always wait, right?), the more the number grows, the more time you’ll need to wait for a result. The time limit: Our code will replicate this process step by step, returning the result. Here’s the code: https://gist.github.com/NicolaM94/5d81f8696c340adc80fcbbff6c957e2d?embedable=true The function accepts two arguments: , the number to multiply as string, and , the other one as integer. First off, we reverse and create two other variables: multiplication n m n , which will hold our new number digit by digit newNumber , which will keep track of the remainder of each operation. The remainder is initially equal to 0. remainder We loop over the reversed number and, at each step: We calculate the of the digit, we are on times the second number and add the remainder. result m We add the last digit of to the ; result newNumber If there is a remainder (when ), we update the variable with the new one. The remainder is composed by the digits of except the last one. If there is no remainder, we set back to 0. result > 9 remainder result remainder This is really important to avoid bringing on remainders of previous multiplications! We keep looping for each digit of and finally, if there is a remainder after the last multiplication, we add it to . After that, we simply return our . reversed newNumber newNumber Calculating the factorial Now that we have a method to multiply numbers of the size we want, we can apply it to calculate our factorials! Here’s the code: https://gist.github.com/NicolaM94/95fee9b37cba1078ec40a100ed525de5?embedable=true It’s just a really: all it does is set a variable , initially equal to the string “1” and then range from 1 to (n+1 really, for Python’s range stops at n-1) to apply our function, each time updating with the new result. wrapper, out n multiplication out For example, here’s the result of 500!: 1220136825991110068701238785423046926253574342803192842192413588385845373153881997605496447502203281863013616477148203584163378722078177200480785205159329285477907571939330603772960859086270429174547882424912726344305670173270769461062802310452644218878789465754777149863494367781037644274033827365397471386477878495438489595537537990423241061271326984327745715546309977202781014561081188373709531016356324432987029563896628911658974769572087926928871281780070265174507768410719624390394322536422605234945850129918571501248706961568141625359056693423813008856249246891564126775654481886506593847951775360894005745238940335798476363944905313062323749066445048824665075946735862074637925184200459369692981022263971952597190945217823331756934581508552332820762820023402626907898342451712006207714640979456116127629145951237229913340169552363850942885592018727433795173014586357570828355780158735432768888680120399882384702151467605445407663535984174430480128938313896881639487469658817504506926365338175055478128640000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 Summing up Since our problem requires us to calculate the sum of the digits of 100!, we also need a function to sum all the digits, but it’s quite simple to build: you can do it directly in your function. Here’s the main: main https://gist.github.com/NicolaM94/46aa1b7cb2d13d2efda3f748c423ae5d?embedable=true#file-factorialcalculator-py As in where the problem was also given by , I won’t give out the solution: you need to build this by yourself, at least, if you want to know it! besides presenting math riddles and problems, . And I love that: they are so committed that publishing results and guides for their problems is actually forbidden (this is one of the first 100 problems, so I understand it is permitted). Given this, . the other article ProjectEuler.net ProjectEuler.net is really a lovely project: their main focus is to provide a tool to start learning, thinking, and reasoning about math I don’t feel it would be fair to just give out the solution to copy-paste and get the achievement Have a nice coding, guys! 😎 Time and complexity It’s not that hard to evaluate the time complexity of this algorithm: the biggest amount of time is spent inside the function. Here, besides allocating some variables and modifying their values, the slowest part is the for loop, which will run times, depending on the length of the number we are multiplying. It is linear, since there are no nested loops to deal with. multiplication len(n) When we wrap this function in though, we must run that function times: so the time complexity becomes quadratic, leaving us with O(n²) t.c. hugeFactorial n Finally, summing up the digits it’s another linear time algorithm, but it’s only invoked after the factorial is calculated: for this reason, it won’t add any complexity, leaving us with a total complexity of . O(n²) Talking about execution time, here’s the time taken by my machine (AMD Ryzen 5, avg. ck. 2700 MHz) to calculate some results: With a simple regression, even if the data is clearly not enough, we can see that our prediction of a O(n²) time complexity holds true: the time taken actually grows like a parabola. I did not have much time to go over 10000!, so if you try it out, please let me know your results! Conclusions Here it is; this was my tackle! In all likelihood, there are better approaches to this kind of problem, but, as always, I wanted to share my try with you guys. What do you think about this? Do you have a better solution or idea? Let me know in the comments! I hope you liked the article and found something interesting in it! If so, please leave a like or share it with someone who could appreciate this kind of problem: that would really make my day! Thanks for reading; as always

Daily Coding Problem: Building an Algorithm to Calculate Huge Factorials!

Off on a Comet! a Journey through Planetary Space by Jules Verne, is part of the HackerNoon Books Series. You can jump to any chapter in this book . THE PROFESSOR’S EXPERIENCES here CHAPTER III. THE PROFESSOR’S EXPERIENCES “Yes, my comet!” repeated the professor, and from time to time he knitted his brows, and looked around him with a defiant air, as though he could not get rid of the impression that someone was laying an unwarranted claim to its proprietorship, or that the individuals before him were intruders upon his own proper domain. But for a considerable while, Servadac, the count, and the lieutenant remained silent and sunk in thought. Here then, at last, was the unriddling of the enigma they had been so long endeavoring to solve; both the hypotheses they had formed in succession had now to give way before the announcement of the real truth. The first supposition, that the rotatory axis of the earth had been subject to some accidental modification, and the conjecture that replaced it, namely, that a certain portion of the terrestrial sphere had been splintered off and carried into space, had both now to yield to the representation that the earth had been grazed by an unknown comet, which had caught up some scattered fragments from its surface, and was bearing them far away into sidereal regions. Unfolded lay the past and the present before them; but this only served to awaken a keener interest about the future. Could the professor throw any light upon that? they longed to inquire, but did not yet venture to ask him. Meanwhile Rosette assumed a pompous professional air, and appeared to be waiting for the entire party to be ceremoniously introduced to him. Nothing unwilling to humor the vanity of the eccentric little man, Servadac proceeded to go through the expected formalities. “Allow me to present to you my excellent friend, the Count Timascheff,” he said. “You are very welcome,” said Rosette, bowing to the count with a smile of condescension. “Although I am not precisely a voluntary resident on your comet, Mr. Professor, I beg to acknowledge your courteous reception,” gravely responded Timascheff. Servadac could not quite conceal his amusement at the count’s irony, but continued, “This is Lieutenant Procope, the officer in command of the Dobryna.” The professor bowed again in frigid dignity. “His yacht has conveyed us right round Gallia,” added the captain. “Round Gallia?” eagerly exclaimed the professor. “Yes, entirely round it,” answered Servadac, and without allowing time for reply, proceeded, “And this is my orderly, Ben Zoof.” “Aide-de-camp to his Excellency the Governor of Gallia,” interposed Ben Zoof himself, anxious to maintain his master’s honor as well as his own. Rosette scarcely bent his head. The rest of the population of the Hive were all presented in succession: the Russian sailors, the Spaniards, young Pablo, and little Nina, on whom the professor, evidently no lover of children, glared fiercely through his formidable spectacles. Isaac Hakkabut, after his introduction, begged to be allowed to ask one question. “How soon may we hope to get back?” he inquired. “Get back!” rejoined Rosette, sharply; “who talks of getting back? We have hardly started yet.” Seeing that the professor was inclined to get angry, Captain Servadac adroitly gave a new turn to the conversation by asking him whether he would gratify them by relating his own recent experiences. The astronomer seemed pleased with the proposal, and at once commenced a verbose and somewhat circumlocutory address, of which the following summary presents the main features. The French Government, being desirous of verifying the measurement already made of the arc of the meridian of Paris, appointed a scientific commission for that purpose. From that commission the name of Palmyrin Rosette was omitted, apparently for no other reason than his personal unpopularity. Furious at the slight, the professor resolved to set to work independently on his own account, and declaring that there were inaccuracies in the previous geodesic operations, he determined to re-examine the results of the last triangulation which had united Formentera to the Spanish coast by a triangle, one of the sides of which measured over a hundred miles, the very operation which had already been so successfully accomplished by Arago and Biot. Accordingly, leaving Paris for the Balearic Isles, he placed his observatory on the highest point of Formentera, and accompanied as he was only by his servant, Joseph, led the life of a recluse. He secured the services of a former assistant, and dispatched him to a high peak on the coast of Spain, where he had to superintend a reverberator, which, with the aid of a glass, could be seen from Formentera. A few books and instruments, and two months’ victuals, was all the baggage he took with him, except an excellent astronomical telescope, which was, indeed, almost part and parcel of himself, and with which he assiduously scanned the heavens, in the sanguine anticipation of making some discovery which would immortalize his name. The task he had undertaken demanded the utmost patience. Night after night, in order to fix the apex of his triangle, he had to linger on the watch for the assistant’s signal-light, but he did not forget that his predecessors, Arago and Biot, had had to wait sixty-one days for a similar purpose. What retarded the work was the dense fog which, it has been already mentioned, at that time enveloped not only that part of Europe, but almost the entire world. Never failing to turn to the best advantage the few intervals when the mist lifted a little, the astronomer would at the same time cast an inquiring glance at the firmament, as he was greatly interested in the revision of the chart of the heavens, in the region contiguous to the constellation Gemini. To the naked eye this constellation consists of only six stars, but through a telescope ten inches in diameter, as many as six thousand are visible. Rosette, however, did not possess a reflector of this magnitude, and was obliged to content himself with the good but comparatively small instrument he had. On one of these occasions, whilst carefully gauging the recesses of Gemini, he espied a bright speck which was unregistered in the chart, and which at first he took for a small star that had escaped being entered in the catalogue. But the observation of a few separate nights soon made it manifest that the star was rapidly changing its position with regard to the adjacent stars, and the astronomer’s heart began to leap at the thought that the renown of the discovery of a new planet would be associated with his name. Redoubling his attention, he soon satisfied himself that what he saw was not a planet; the rapidity of its displacement rather forced him to the conjecture that it must be a comet, and this opinion was soon strengthened by the appearance of a coma, and subsequently confirmed, as the body approached the sun, by the development of a tail. A comet! The discovery was fatal to all further progress in the triangulation. However conscientiously the assistant on the Spanish coast might look to the kindling of the beacon, Rosette had no glances to spare for that direction; he had no eyes except for the one object of his notice, no thoughts apart from that one quarter of the firmament. A comet! No time must be lost in calculating its elements. Now, in order to calculate the elements of a comet, it is always deemed the safest mode of procedure to assume the orbit to be a parabola. Ordinarily, comets are conspicuous at their perihelia, as being their shortest distances from the sun, which is the focus of their orbit, and inasmuch as a parabola is but an ellipse with its axis indefinitely produced, for some short portion of its pathway the orbit may be indifferently considered either one or the other; but in this particular case the professor was right in adopting the supposition of its being parabolic. Just as in a circle, it is necessary to know three points to determine the circumference; so in ascertaining the elements of a comet, three different positions must be observed before what astronomers call its “ephemeris” can be established. But Professor Rosette did not content himself with three positions; taking advantage of every rift in the fog he made ten, twenty, thirty observations both in right ascension and in declination, and succeeded in working out with the most minute accuracy the five elements of the comet which was evidently advancing with astounding rapidity towards the earth. These elements were: The position of the ascending node, or the point where the comet crossed the terrestrial orbit. These two elements being obtained, the position in space of the comet’s orbit was determined. The direction of the axis major of the orbit, which was found by calculating the longitude of the comet’s perihelion. The perihelion distance from the sun, which settled the precise form of the parabola. The motion of the comet, as being retrograde, or, unlike the planets, from east to west. Rosette thus found himself able to calculate the date at which the comet would reach its perihelion, and, overjoyed at his discovery, without thinking of calling it Palmyra or Rosette, after his own name, he resolved that it should be known as Gallia. His next business was to draw up a formal report. Not only did he at once recognize that a collision with the earth was possible, but he soon foresaw that it was inevitable, and that it must happen on the night of the 31st of December; moreover, as the bodies were moving in opposite directions, the shock could hardly fail to be violent. To say that he was elated at the prospect was far below the truth; his delight amounted almost to delirium. Anyone else would have hurried from the solitude of Formentera in sheer fright; but, without communicating a word of his startling discovery, he remained resolutely at his post. From occasional newspapers which he had received, he had learnt that fogs, dense as ever, continued to envelop both hemispheres, so that he was assured that the existence of the comet was utterly unknown elsewhere; and the ignorance of the world as to the peril that threatened it averted the panic that would have followed the publication of the facts, and left the philosopher of Formentera in sole possession of the great secret. He clung to his post with the greater persistency, because his calculations had led him to the conclusion that the comet would strike the earth somewhere to the south of Algeria, and as it had a solid nucleus, he felt sure that, as he expressed it, the effect would be “unique,” and he was anxious to be in the vicinity. The shock came, and with it the results already recorded. Palmyrin Rosette was suddenly separated from his servant Joseph, and when, after a long period of unconsciousness, he came to himself, he found that he was the solitary occupant of the only fragment that survived of the Balearic Archipelago. Such was the substance of the narrative which the professor gave with sundry repetitions and digressions; while he was giving it, he frequently paused and frowned as if irritated in a way that seemed by no means justified by the patient and good-humored demeanor of his audience. “But now, gentlemen,” added the professor, “I must tell you something more. Important changes have resulted from the collision; the cardinal points have been displaced; gravity has been diminished: not that I ever supposed for a minute, as you did, that I was still upon the earth. No! the earth, attended by her moon, continued to rotate along her proper orbit. But we, gentlemen, have nothing to complain of; our destiny might have been far worse; we might all have been crushed to death, or the comet might have remained in adhesion to the earth; and in neither of these cases should we have had the satisfaction of making this marvelous excursion through untraversed solar regions. No, gentlemen, I repeat it, we have nothing to regret.” And as the professor spoke, he seemed to kindle with the emotion of such supreme contentment that no one had the heart to gainsay his assertion. Ben Zoof alone ventured an unlucky remark to the effect that if the comet had happened to strike against Montmartre, instead of a bit of Africa, it would have met with some resistance. “Pshaw!” said Rosette, disdainfully. “A mole-hill like Montmartre would have been ground to powder in a moment.” “Mole-hill!” exclaimed Ben Zoof, stung to the quick. “I can tell you it would have caught up your bit of a comet and worn it like a feather in a cap.” The professor looked angry, and Servadac having imposed silence upon his orderly, explained the worthy soldier’s sensitiveness on all that concerned Montmartre. Always obedient to his master, Ben Zoof held his tongue; but he felt that he could never forgive the slight that had been cast upon his beloved home. It was now all-important to learn whether the astronomer had been able to continue his observations, and whether he had learned sufficient of Gallia’s path through space to make him competent to determine, at least approximately, the period of its revolution round the sun. With as much tact and caution as he could, Lieutenant Procope endeavored to intimate the general desire for some information on this point. “Before the shock, sir,” answered the professor, “I had conclusively demonstrated the path of the comet; but, in consequence of the modifications which that shock has entailed upon my comet’s orbit, I have been compelled entirely to recommence my calculations.” The lieutenant looked disappointed. “Although the orbit of the earth was unaltered,” continued the professor, “the result of the collision was the projection of the comet into a new orbit altogether.” “And may I ask,” said Procope, deferentially, “whether you have got the elements of the fresh orbit?” “Yes.” “Then perhaps you know—” “I know this, sir, that at 47 minutes 35.6 seconds after two o’clock on the morning of the 1st of January last, Gallia, in passing its ascending node, came in contact with the earth; that on the 10th of January it crossed the orbit of Venus; that it reached its perihelion on the 15th; that it re-crossed the orbit of Venus; that on the 1st of February it passed its descending node; on the 13th crossed the orbit of Mars; entered the zone of the telescopic planets on the 10th of March, and, attracting Nerina, carried it off as a satellite.” Servadac interposed: “We are already acquainted with well-nigh all these extraordinary facts; many of them, moreover, we have learned from documents which we have picked up, and which, although unsigned, we cannot entertain a doubt have originated with you.” Professor Rosette drew himself up proudly and said: “Of course, they originated with me. I sent them off by hundreds. From whom else could they come?” “From no one but yourself, certainly,” rejoined the count, with grave politeness. Hitherto the conversation had thrown no light upon the future movements of Gallia, and Rosette was disposed apparently to evade, or at least to postpone, the subject. When, therefore, Lieutenant Procope was about to press his inquiries in a more categorical form, Servadac, thinking it advisable not prematurely to press the little savant too far, interrupted him by asking the professor how he accounted for the earth having suffered so little from such a formidable concussion. “I account for it in this way,” answered Rosette: “the earth was traveling at the rate of 28,000 leagues an hour, and Gallia at the rate of 57,000 leagues an hour, therefore the result was the same as though a train rushing along at a speed of about 86,000 leagues an hour had suddenly encountered some obstacle. The nucleus of the comet, being excessively hard, has done exactly what a ball would do fired with that velocity close to a pane of glass. It has crossed the earth without cracking it.” “It is possible you may be right,” said Servadac, thoughtfully. “Right! of course I am right!” replied the snappish professor. Soon, however, recovering his equanimity, he continued: “It is fortunate that the earth was only touched obliquely; if the comet had impinged perpendicularly, it must have plowed its way deep below the surface, and the disasters it might have caused are beyond reckoning. Perhaps,” he added, with a smile, “even Montmartre might not have survived the calamity.” “Sir!” shouted Ben Zoof, quite unable to bear the unprovoked attack. “Quiet, Ben Zoof!” said Servadac sternly. Fortunately for the sake of peace, Isaac Hakkabut, who at length was beginning to realize something of the true condition of things, came forward at this moment, and in a voice trembling with eagerness, implored the professor to tell him when they would all be back again upon the earth. “Are you in a great hurry?” asked the professor coolly. The Jew was about to speak again, when Captain Servadac interposed: “Allow me to say that, in somewhat more scientific terms, I was about to ask you the same question. Did I not understand you to say that, as the consequence of the collision, the character of the comet’s orbit has been changed?” “You did, sir.” “Did you imply that the orbit has ceased to be a parabola?” “Just so.” “Is it then an hyperbola? and are we to be carried on far and away into remote distance, and never, never to return?” “I did not say an hyperbola.” “And is it not?” “It is not.” “Then it must be an ellipse?” “Yes.” “And does its plane coincide with the plane of the earth?” “Yes.” “Then it must be a periodic comet?” “It is.” Servadac involuntarily raised a ringing shout of joy that echoed again along the gallery. “Yes,” continued the professor, “Gallia is a periodic comet, and allowing for the perturbations to which it is liable from the attraction of Mars and Jupiter and Saturn, it will return to the earth again in two years precisely.” “You mean that in two years after the first shock, Gallia will meet the earth at the same point as they met before?” said Lieutenant Procope. “I am afraid so,” said Rosette. “Why afraid?” “Because we are doing exceedingly well as we are.” The professor stamped his foot upon the ground, by way of emphasis, and added, “If I had my will, Gallia should never return to the earth again!” About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books. This book is part of the public domain. Jules Verne (1998). Off on a Comet! a Journey through Planetary Space. Urbana, Illinois: Project Gutenberg. Retrieved October 2022 https://www.gutenberg.org/cache/epub/1353/pg1353-images.html This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at , located at . www.gutenberg.org https://www.gutenberg.org/policy/license.html

THE PROFESSOR’S EXPERIENCES

Off on a Comet! a Journey through Planetary Space by Jules Verne, is part of the HackerNoon Books Series. You can jump to any chapter in this book . THE ASTRONOMER here CHAPTER I. THE ASTRONOMER By the return of the expedition, conveying its contribution from Formentera, the known population of Gallia was raised to a total of thirty-six. On learning the details of his friends’ discoveries, Count Timascheff did not hesitate in believing that the exhausted individual who was lying before him was the author alike of the two unsigned documents picked up at sea, and of the third statement so recently brought to hand by the carrier-pigeon. Manifestly, he had arrived at some knowledge of Gallia’s movements: he had estimated her distance from the sun; he had calculated the diminution of her tangential speed; but there was nothing to show that he had arrived at the conclusions which were of the most paramount interest to them all. Had he ascertained the true character of her orbit? had he established any data from which it would be possible to reckon what time must elapse before she would again approach the earth? The only intelligible words which the astronomer had uttered had been, “My comet!” To what could the exclamation refer? Was it to be conjectured that a fragment of the earth had been chipped off by the collision of a comet? and if so, was it implied that the name of the comet itself was Gallia, and were they mistaken in supposing that such was the name given by the to the little world that had been so suddenly launched into space? Again and again they discussed these questions; but no satisfactory answer could be found. The only man who was able to throw any light upon the subject was lying amongst them in an unconscious and half-dying condition. savant Apart from motives of humanity, motives of self-interest made it a matter of the deepest concern to restore animation to that senseless form. Ben Zoof, after making the encouraging remark that have as many lives as a cat, proceeded, with Negrete’s assistance, to give the body such a vigorous rubbing as would have threatened serious injury to any ordinary mortal, whilst they administered cordials and restoratives from the medical stores powerful enough, one might think, to rouse the very dead. savants Dobryna’s Meanwhile the captain was racking his brain in his exertions to recall what were the circumstances of his previous acquaintance with the Frenchman upon whose features he was gazing; he only grew more and more convinced that he had once been familiar with them. Perhaps it was not altogether surprising that he had almost forgotten him; he had never seen him since the days of his youth, that time of life which, with a certain show of justice, has been termed the age of ingratitude; for, in point of fact, the astronomer was none other than Professor Palmyrin Rosette, Servadac’s old science-master at the Lycee Charlemagne. After completing his year of elementary studies, Hector Servadac had entered the school at Saint Cyr, and from that time he and his former tutor had never met, so that naturally they would well-nigh pass from each other’s recollection. One thing, however, on the other hand, might conduce to a mutual and permanent impression on their memories; during the year at the Lycee, young Servadac, never of a very studious turn of mind, had contrived, as the ringleader of a set of like caliber as himself, to lead the poor professor a life of perpetual torment. On the discovery of each delinquency he would fume and rage in a manner that was a source of unbounded delight to his audience. Two years after Servadac left the Lycee, Professor Rosette had thrown up all educational employment in order that he might devote himself entirely to the study of astronomy. He endeavored to obtain a post at the Observatory, but his ungenial character was so well known in scientific circles that he failed in his application; however, having some small private means, he determined on his own account to carry on his researches without any official salary. He had really considerable genius for the science that he had adopted; besides discovering three of the latest of the telescopic planets, he had worked out the elements of the three hundred and twenty-fifth comet in the catalogue; but his chief delight was to criticize the publications of other astronomers, and he was never better pleased than when he detected a flaw in their reckonings. When Ben Zoof and Negrete had extricated their patient from the envelope of furs in which he had been wrapped by Servadac and the lieutenant, they found themselves face to face with a shrivelled little man, about five feet two inches high, with a round bald head, smooth and shiny as an ostrich’s egg, no beard unless the unshorn growth of a week could be so described, and a long hooked nose that supported a huge pair of spectacles such as with many near-sighted people seems to have become a part of their individuality. His nervous system was remarkably developed, and his body might not inaptly be compared to one of the Rhumkorff’s bobbins of which the thread, several hundred yards in length, is permeated throughout by electric fluid. But whatever he was, his life, if possible, must be preserved. When he had been partially divested of his clothing, his heart was found to be still beating, though very feebly. Asserting that while there was life there was hope, Ben Zoof recommenced his friction with more vigor than ever. When the rubbing had been continued without a moment’s intermission for the best part of half an hour, the astronomer heaved a faint sigh, which ere long was followed by another and another. He half opened his eyes, closed them again, then opened them completely, but without exhibiting any consciousness whatever of his situation. A few words seemed to escape his lips, but they were quite unintelligible. Presently he raised his right hand to his forehead as though instinctively feeling for something that was missing; then, all of a sudden, his features became contracted, his face flushed with apparent irritation, and he exclaimed fretfully, “My spectacles!—where are my spectacles?” In order to facilitate his operations, Ben Zoof had removed the spectacles in spite of the tenacity with which they seemed to adhere to the temples of his patient; but he now rapidly brought them back and readjusted them as best he could to what seemed to be their natural position on the aquiline nose. The professor heaved a long sigh of relief, and once more closed his eyes. Before long the astronomer roused himself a little more, and glanced inquiringly about him, but soon relapsed into his comatose condition. When next he opened his eyes, Captain Servadac happened to be bending down closely over him, examining his features with curious scrutiny. The old man darted an angry look at him through the spectacles, and said sharply, “Servadac, five hundred lines to-morrow!” It was an echo of days of old. The words were few, but they were enough to recall the identity which Servadac was trying to make out. “Is it possible?” he exclaimed. “Here is my old tutor, Mr. Rosette, in very flesh and blood.” “Can’t say much for the flesh,” muttered Ben Zoof. The old man had again fallen back into a torpid slumber. Ben Zoof continued, “His sleep is getting more composed. Let him alone; he will come round yet. Haven’t I heard of men more dried up than he is, being brought all the way from Egypt in cases covered with pictures?” “You idiot!—those were mummies; they had been dead for ages.” Ben Zoof did not answer a word. He went on preparing a warm bed, into which he managed to remove his patient, who soon fell into a calm and natural sleep. Too impatient to await the awakening of the astronomer and to hear what representations he had to make, Servadac, the count, and the lieutenant, constituting themselves what might be designated “the Academy of Sciences” of the colony, spent the whole of the remainder of the day in starting and discussing the wildest conjectures about their situation. The hypothesis, to which they had now accustomed themselves for so long, that a new asteroid had been formed by a fracture of the earth’s surface, seemed to fall to the ground when they found that Professor Palmyrin Rosette had associated the name of Gallia, not with their present home, but with what he called “my comet”; and that theory being abandoned, they were driven to make the most improbable speculations to replace it. Alluding to Rosette, Servadac took care to inform his companions that, although the professor was always eccentric, and at times very irascible, yet he was really exceedingly good-hearted; his bark was worse than his bite; and if suffered to take their course without observation, his outbreaks of ill-temper seldom lasted long. “We will certainly do our best to get on with him,” said the count. “He is no doubt the author of the papers, and we must hope that he will be able to give us some valuable information.” “Beyond a question the documents have originated with him,” assented the lieutenant. “Gallia was the word written at the top of every one of them, and Gallia was the first word uttered by him in our hearing.” The astronomer slept on. Meanwhile, the three together had no hesitation in examining his papers, and scrutinizing the figures on his extemporized blackboard. The handwriting corresponded with that of the papers already received; the blackboard was covered with algebraical symbols traced in chalk, which they were careful not to obliterate; and the papers, which consisted for the most part of detached scraps, presented a perfect wilderness of geometrical figures, conic sections of every variety being repeated in countless profusion. Lieutenant Procope pointed out that these curves evidently had reference to the orbits of comets, which are variously parabolic, hyperbolic, or elliptic. If either of the first two, the comet, after once appearing within the range of terrestrial vision, would vanish forever in the outlying regions of space; if the last, it would be sure, sooner or later, after some periodic interval, to return. From the appearance of his papers, then, it seemed probable that the astronomer, during his sojourn at Formentera, had been devoting himself to the study of cometary orbits; and as calculations of this kind are ordinarily based upon the assumption that the orbit is a parabola, it was not unlikely that he had been endeavoring to trace the path of some particular comet. prima facie “I wonder whether these calculations were made before or after the 1st of January; it makes all the difference,” said Lieutenant Procope. “We must bide our time and hear,” replied the count. Servadac paced restlessly up and down. “I would give a month of my life,” he cried, impetuously, “for every hour that the old fellow goes sleeping on.” “You might be making a bad bargain,” said Procope, smiling. “Perhaps after all the comet has had nothing to do with the convulsion that we have experienced.” “Nonsense!” exclaimed the captain; “I know better than that, and so do you. Is it not as clear as daylight that the earth and this comet have been in collision, and the result has been that our little world has been split off and sent flying far into space?” Count Timascheff and the lieutenant looked at each other in silence. “I do not deny your theory,” said Procope after a while. “If it be correct, I suppose we must conclude that the enormous disc we observed on the night of the catastrophe was the comet itself; and the velocity with which it was traveling must have been so great that it was hardly arrested at all by the attraction of the earth.” “Plausible enough,” answered Count Timascheff; “and it is to this comet that our scientific friend here has given the name of Gallia.” It still remained a puzzle to them all why the astronomer should apparently be interested in the comet so much more than in the new little world in which their strange lot was cast. “Can you explain this?” asked the count. “There is no accounting for the freaks of philosophers, you know,” said Servadac; “and have I not told you that this philosopher in particular is one of the most eccentric beings in creation?” “Besides,” added the lieutenant, “it is exceedingly likely that his observations had been going on for some considerable period before the convulsion happened.” Thus, the general conclusion arrived at by the Gallian Academy of Science was this: That on the night of the 31st of December, a comet, crossing the ecliptic, had come into collision with the earth, and that the violence of the shock had separated a huge fragment from the globe, which fragment from that date had been traversing the remote inter-planetary regions. Palmyrin Rosette would doubtless confirm their solution of the phenomenon. About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books. This book is part of the public domain. Jules Verne (1998). Off on a Comet! a Journey through Planetary Space. Urbana, Illinois: Project Gutenberg. Retrieved October 2022 https://www.gutenberg.org/cache/epub/1353/pg1353-images.html This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at , located at . www.gutenberg.org https://www.gutenberg.org/policy/license.html

THE ASTRONOMER

Head and Shoulders

The Offshore Pirate

Do you have complete confidence in the responses provided by ChatGPT or Google Bard when it comes to solving mathematical problems? Well, let me present you with some important findings. I conducted a series of inquiries to assess the accuracy of AI solutions. I began with simple quadratic equations and gradually progressed to more complex problems, and I aimed to evaluate the problem-solving capabilities of AI. Considering OpenAI's caution regarding potentially incorrect or harmful responses from ChatGPT, my article will contribute to the improvement and proper training of AI models in the future. It's worth noting that Google Bard is still in its experimental phase and has shown comparatively more errors than ChatGPT. So, let’s get started with a few examples. First example: Initially, I asked ChatGPT to solve the following problem to solve: “Check whether  the roots of x obtained from the equation 5^(x*x)*5^x-15625=0 satisfies other equations 9^(x*x)*9^x-81=0 or not.” ChatGPT: ⚠️Check carefully for x = -3, it calculated the value 531360, which is not equal to zero, but it declares that the equation is satisfied, amazing! Now let’s head to Google Bard. I asked the same question: ⚠️Oh, no! It incorrectly calculated the roots. Check the code snippets. ✅Both roots of x are obtained from the first equation, i.e., 2 and -3 does not satisfy by the second equation because for x = 2, it gives 531360, and for x = -3, it gives 531360 but let’s see how ChatGPT and Google Bard have responded. They are not as mature as you think in terms of mathematical problem-solving. Second example: Here, I asked the 12th grade’s mathematical problem to test AI responses. “If logx(1 / 8) = -3 / 4, than what is x? (x is base of log)” ChatGPT: ⚠️Here, it creates a mistake in the highlighted area as the correct calculation should be (½)^4. Google Bard: ⚠️It has created a mistake while calculating the third step in the code snippet. ✅The correct calculation gives us a value of x 16, but AI models failed to calculate it. Let’s see: The correct solution: Rewriting the equation in the exponential form gives: x^(- 3 / 4) = 1 / 8 x= (1 / 8)^(- 4 / 3) x= 2^4 = 16 Example 3: Here, I asked the B.Sc. level mathematical problem to test AI responses. “The area bounded by the parabola y^2=4px and the line x = a revolves about the x-axis. Find the exact value of the volume generated.” ChatGPT: ⚠️Here, the formula used by ChatGPT to calculate V (in the highlighted area) is incorrect as it has to be V = ∫πy²dx; within limits 0 to a. Google Bard: ⚠️Here, Google Bard has also generated an incorrect answer (8/3)*a^4. ✅The correct solution to the question is, First, volume (V) = ∫πy²dx ; within limits a to 0 = π ∫4ax dx = 4aπ [x²/2]; within the limit 0 to a = 4aπ.(a²/2) = 2πa³ Example 3: Here, I ask AI bots to solve another math problem to check their responses: “A semicircular piece of paper is folded to make a cone with the centre of the semicircle as the apex. The half-angle of the resulting cone would be:” ChatGPT: ⚠️Here, ChatGPT created the mistakes as the circumference of the semicircle “πr” should be equal to the circumference of the cone “2πl*sin(θ)”.\ Google Bard: ⚠️Surprisingly, Google Bard has found 90 degrees as the half angle of the resulting cone with the incorrect reference. ✅ The correct answer in this case: Let’s say R is the radius of the semicircle, then the slant height of the cone made from it is also R. If r is the radius of the base of the cone, then, 2πr = πR So, r = R/2 Let A be the half angle of the cone, then, SinA = (R/2)/R A = 30 degrees Despite its several mistakes in the calculations, ChatGPT has genuinely solved some integration-based problems, for example: Question: Solve ∫xtan^2(x)dx So, all the above showcases indicate that AI chatbots are still in the experimental phases to solve mathematical problems. As they are backed by huge datasets, chatbots should be properly trained to perform mathematical calculations without any errors. In the above examples, ChatGPT has exhibited some minor to severe mistakes. If you receive the incorrect solutions or information for your problems, I ask you to “Regenerate” responses until the Chatbots give you accurate solutions; else, you can suggest the correct solutions so that next time bots may offer you the correct solutions. In the cases of the above mistakes, I provided detailed feedback to ChatGPT so that in the next attempts, I received the correct solutions to the above problems. You can also give it a try to test whether it shares accurate information or the same recurring mistakes. Conclusion: Chatbots do not share accurate information or the solutions for your mathematical problems. Are you leaving your homework to AI chatbots to solve? Be aware they are not mature enough in mathematical problem-solving; however, the bots are slowly but steadily improving their responses. If the training data of the bots are incorrect, then they can not give you precise solutions. In addition, AI chatbots like ChatGPT and Google Bard may not receive extensive training specifically in mathematics, so they still can’t offer knowledge with advanced mathematical reasoning skills. So, if you are a student, teacher, or expert in the related field and notice the mistakes, help AI models by offering feedback so that a huge number of users will get benefits in the future.

Weakness of AI Models in Terms of Mathematical Problem Solving

All unite here in this paradise of the Bembex

Edmund Halley, a soap-boiler

THE BEMBEX

THE LOCUSTS: THEIR EGGS

More Hunting Wasps by Jean-Henri Fabre, is part of the HackerNoon Books Series. You can jump to any chapter in this book here . A DIG AT THE EVOLUTIONISTS CHAPTER 8. A DIG AT THE EVOLUTIONISTS. To rear a caterpillar-eater on a skewerful of Spiders is a very innocent thing, unlikely to compromise the security of the State; it is also a very childish thing, as I hasten to confess, and worthy of the schoolboy who, in the mysteries of his desk, seeks as best he may some diversion from the fascinations of his exercise in composition. And I should not have undertaken these investigations, still less should I have spoken them, not without some satisfaction, if I had not discerned, in the results obtained in my refectory, a certain philosophic import, involving, so it seemed to me, the evolutionary theory. It is assuredly a majestic enterprise, commensurate with man's immense ambitions, to seek to pour the universe into the mould of a formula and submit every reality to the standard of reason. The geometrician proceeds in this manner: he defines the cone, an ideal conception; then he intersects it by a plane. The conic section is submitted to algebra, an obstetrical appliance which brings forth the equation; and behold, entreated now in one direction, now in another, the womb of the formula gives birth to the ellipse, the hyperbola, the parabola, their foci, their radius vectors, their tangents, their normals, their conjugate axes, their asymptotes and the rest. It is magnificent, so much so that you are overcome by enthusiasm, even when you are twenty years old, an age hardly adapted to the austerities of mathematics. It is superb. You feel as if you were witnessing the creation of a world. As a matter of fact, you are merely observing the same idea from different points of view, which are illumined by the successive phases of the transformed formula. All that algebra unfolds for our benefit was contained in the definition of the cone, but it was contained as a germ, under latent forms which the magic of the calculus converts into explicit forms. The gross value which our mind confided to the equation it returns to us, without loss or gain, in coins stamped with every sort of effigy. And here precisely is that which constitutes the inflexible rigour of the calculus, the luminous certainty before which every cultivated mind is forced to bow. Algebra is the oracle of the absolute truth, because it reveals nothing but what the mind had hidden in it under an amalgam of symbols. We put 2 and 2 into the machine; the rollers work and show us 4. That is all. But to this calculus, all-powerful so long as it does not leave the domain of the ideal, let us submit a very modest reality: the fall of a grain of sand, the pendular movement of a hanging body. The machine no longer works, or does so only by suppressing almost everything that is real. It must have an ideal material point, an ideal rigid thread, an ideal point of suspension; and then the pendular movement is translated by a formula. But the problem defies all the artifices of analysis if the oscillating body is a real body, endowed with volume and friction; if the suspensory thread is a real thread, endowed with weight and flexibility; if the point of support is a real point, endowed with resistance and capable of deflection. So with other problems, however simple. The exact reality escapes the formula. Yes, it would be a fine thing to put the world into an equation, to assume as the first principle a cell filled with albumen and by transformation after transformation to discover life under its thousand aspects as the geometrician discovers the ellipse and the other curves by examining his conic section. Yes, it would be magnificent and enough to add a cubit to our stature. Alas, how greatly must we abate our pretensions! The reality is beyond our reach when it is only a matter of following a grain of dust in its fall; and we would undertake to ascend the river of life and trace it to its source! The problem is a more arduous one than that which algebra declines to solve. There are formidable unknown quantities here, more difficult to decipher than the resistances, the deflections and the frictions of the pendulum. Let us eliminate them, that we may more easily propound the theory. Very well; but then my confidence in this natural history which repudiates nature and gives ideal conceptions precedence over real facts is shaken. So, without seeking the opportunity, which is not my business, I take it when it presents itself; I examine the theory of evolution from every side; and, as that which I have been assured is the majestic dome of a monument capable of defying the ages appears to me to be no more than a bladder, I irreverently dig my pin into it. Here is the latest dig. Adaptability to a varied diet is an element of well-being in the animal, a factor of prime importance for the extension and predominance of its race in the bitter struggle for life. The most unfortunate species would be that which depended for its existence on a diet so exclusive that no other could replace it. What would become of the Swallow if he required, in order to live, one particular Gnat, a single Gnat, always the same? When once this Gnat had disappeared—and the life of the Mosquito is not a long one—the bird would die of starvation. Fortunately for himself and for the happiness of our homes, the Swallow gulps them all down indiscriminately, together with a host of other insects that perform aerial ballets. What would become of the Lark were his gizzard able to digest only one seed, invariably the same? When the season for this seed was over—and the season is always a short one—the haunter of the furrows would perish. Is not man's complaisant stomach, adapted to the largest variety of nourishment, one of his great zoological privileges? He is thus rendered independent of climates, seasons and latitudes. And the Dog: how is it that of all the domestic animals he alone is able to accompany us everywhere, even on the most arduous expeditions? The Dog again is omnivorous and therefore a cosmopolitan. The discovery of a new dish, said Brillat-Savarin, is of greater importance to humanity than the discovery of a new planet. The aphorism is nearer to the truth than it appears to be in its humorous form. Certainly the man who was the first to think of crushing wheat, kneading flour and cooking the paste between two hot stones was more deserving than the discoverer of the two-hundredth asteroid. The invention of the potato is certainly as valuable as that of Neptune, glorious as the latter was. All that increases our alimentary resources is a discovery of the first merit. And what is true of man cannot be other than true of animals. The world belongs to the stomach which is independent of specialities. This truth is of the kind that has only to be stated to be proved. Let us now return to our insects. If I am to believe the evolutionists, the various game-hunting Wasps are descended from a small number of types, which are themselves derived, by an incalculable number of concatenations, from a few amoebae, a few monera and lastly from the first clot of protoplasm which was casually condensed. Let us not go back as far as that; let us not plunge into the fogs where illusion and error too easily find a lurking-place. Let us consider a subject with exact limits to it; this is the only way to understand one another. The Sphegidae are descended from a single type, which itself was already a highly-developed descendant and, like its successors, fed its family on prey. The close similarity in form, in colouring and, above all, in habits seem to refer the Tachytes to the same origin. This is ample; let us be satisfied with it. And now please tell me, what did this prototype of the Sphegidae hunt? Was its diet varied or uniform? If we cannot decide, let us examine the two cases. The diet was varied. I heartily congratulate the first born of the Sphex-wasps. She enjoyed the most favourable conditions for leaving a prosperous offspring. Accommodating herself to any kind of prey not disproportionate to her strength, she avoided the dearth of a given species of game at this or that time and in this or that place; she always found the wherewithal to endow her family magnificently, they being, for that matter, fairly indifferent to the nature of the victuals, provided that these consisted of fresh insect-flesh, as the tastes of their cousins many times removed prove to this day. This matriarch of the Sphex clan bore within herself the best chances of assuring victory to her offspring in that pitiless fight for existence which eliminates the weakly and incapable and allows none but the strong and industrious to survive; she possessed an aptitude of great value which atavism could not fail to hand down and which her descendants, who are greatly interested in preserving this magnificent inheritance, must have permanently adopted and even accentuated from one generation to the next, from one branch, one offshoot, to another. Instead of this unscrupulously omnivorous race, levying booty upon every kind of game, to its very great advantage, what do we see to-day? Each Sphex is stupidly limited to an unvarying diet; she hunts only one kind of prey, though her larva accepts them all. One will have nothing but the Ephippiger and must have a female at that; another will have nothing but the Cricket. This one hunts the Locust and nothing else; that one the Mantis and the Empusa. Yet another is addicted to the Grey Worm and another to the Looper. Fools! How great was your mistake in allowing the wise eclecticism of your ancestress, whose relics now repose in the hard mud of some lacustrian stratum, to become obsolete! How much better would things be for you and yours! Abundance is assured; painful and often fruitless searches are avoided; the larder is crammed without being subject to the accidents of time, place and climate. When Ephippigers run short, you fall back upon Crickets; when there are no Crickets, you capture Grasshoppers. But no, my beautiful Sphex-wasps, you were not such fools as that. If in our days you are each confined to a standing family-dish, it is because your ancestress of the lacustrian schists never taught you variety. Could she have taught you uniformity? Let us suppose that the Sphex of antiquity, a novice in the gastronomic art, prepared her potted meats with a single kind of game, no matter what. It was then her descendants who, subdivided into groups and constituted into so many distinct species by the slow travail of the centuries, realized that in addition to the ancestral fare there existed a host of other foods. Tradition being abandoned, there was nothing to guide their choice. They therefore tried a bit of everything in the way of insect game, at hap-hazard; and each time the larva, whose tastes alone had to be consulted, was satisfied with the food supplied, as it is to-day in the refectory provisioned by my care. Every attempt led to the invention of a new dish, an important event, according to the masters, an inestimable resource for the family, who were thereby delivered from the menace of death and enabled to thrive over large areas whence the absence or rarity of a uniform game would have excluded it. And, after making use of a host of different viands in order to attain the culinary variety which is to-day adopted by the whole of the Sphex nation, lo and behold, each species confines itself to a single sort of game, outside which every specimen is obstinately refused, not at table, of course, but in the hunting-field! By your experiments, from age to age, to have discovered variety in diet; to have practised it, to the great advantage of your race, and to end up with uniformity, the cause of decadence; to have known the excellent and to repudiate it for the middling: oh, my Sphex-wasps, it would be stupid if the theory of evolution were correct! To avoid insulting you and also from respect for common sense, I prefer therefore to believe that, if in our days you confine your hunting to a single kind of game, it is because you have never known any other. I prefer to believe that your common ancestress, your precursor, whether her tastes were simple or complex, is a pure chimera, for, if they were any relationship between you, having tested everything in order to arrive at the actual food of each species, having eaten everything and found it grateful to the stomach, you would now, from first to last, be unprejudiced consumers, omnivorous progressives. I prefer to believe, in short, that the theory of evolution is powerless to explain your diet. This is the conclusion drawn from the dining-room installed in my old sardine-box. About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books. This book is part of the public domain. Jean-Henri Fabre (2002). More Hunting Wasps. Urbana, Illinois: Project Gutenberg. 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