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Lucidity has raised $10 million since its 2017 launch. Part of the latest financing came from promissory notes that converted with this round of funding, a spokesperson for the company said. The company didn’t name the investors who participated in the latest funding round.

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As a seasoned developer, ensuring reliability forms the bedrock of every project I undertake. While testing remains mandatory, traditional methods often fall short when dealing with intricate work and evolving requirements. Laravel To conquer these challenges, I've embraced contract-based testing to validate new features and obliterate bugs before product delivery. By establishing intended behaviors and expected outcomes, contracts ensure my code operates as intended, not just for the present but also for future iterations. This dedication to quality has reaped substantial rewards for clients. Project budgets remain stable as rework diminishes. Users revel in seamless experiences and gain confidence, all while maintaining flexibility to adjust the scope down the line. This post details how contract testing elevated one specific application. From the setup process to ongoing maintenance, witness how prioritizing quality assurance at the onset improves deliverables and fortifies client relations. Once you grasp the process, you'll feel significantly more assured before you Hire Laravel Developers. Deciphering Contract Tests stands as an approach that verifies public APIs by stipulating expected application behaviors instead of delving into implementation specifics. Unlike unit or integration tests, contracts ascertain that the code performs as intended now and has the flexibility to handle unforeseen changes in the future. Contract testing Contracts primarily outline the public methods and properties of a class, essentially establishing an "agreement" with external users. Subsequent tests then simulate the usage scenarios of this public API and make relevant assertions. This clear separation keeps the "contract" distinct from its internal coding. Several key benefits include: Emphasis on essential behaviors: Tests validate pivotal expected inputs/outputs rather than exhaustively testing every code path, focusing on vital use cases. Enduring stability: Contracts serve as both documentation and shields against regression stemming from code changes. Tests continue to pass as code evolves provided the output adheres to the contract. Reader-centric documentation: Contracts serve not just as tests but also as living documentation that developers can refer to in order to comprehend the proper usage of classes without delving into code intricacies. A simple example contract class validating a common "CountAll" method is portrayed below: interface PostRepositoryContract { public function countAll(): int; } class PostRepositoryContractTest extends TestCase { public function testCountAllReturnsInteger() { $mock = Mockery::mock(PostRepositoryContract::class); $count = $mock->countAll(); $this->assertInternalType('integer', $count) } } This contract specifically validates that the method returns an integer, without concerning itself with how the counting process is implemented, thus ensuring the stability of tests over time. Installing Contract Tests in Laravel The setup of contract tests in Laravel involves a few simple steps to configure your environment. Installing Required Packages The principal package required is PHPUnit. Integrate it as a development dependency by running composer require --dev phpunit/phpunit. Additionally, consider auxiliary packages such as Mockery for generating test doubles. Utilize it by executing composer require --dev mockery/mockery. Configuring the Testing Environment Add phpunit.xml to your project's root, outlining the basic configuration. This file directs PHPUnit to locate your tests. <?xml version="1.0" encoding="UTF-8"?> <phpunit> <testsuites> <testsuite name="App Tests"> <directory>tests</directory> </testsuite> </testsuites> </phpunit> Generating a Preliminary Test Template Create a tests directory and a sample test file like Feature/UserTest.php. Import the necessary classes and traits to form the foundation for your tests. <?php use PHPUnit\Framework\TestCase; class FeatureTest extends TestCase { public function setUp(): void { //... } public function testExample() { } } Your environment is now primed to initiate focused contract tests! A Real-Life Application of Contract Testing To showcase contract testing in action, let's examine a real-world scenario. Selecting a Class for Testing In this demonstration, let's focus on a fundamental Repository class that engages with the database, such as a PostRepository. This class is pivotal for retrieving, creating, and updating posts, forming the core functionality of our application. Outlining the Public API Contract First and foremost, define the public methods and properties that the repository exposes using an interface: interface PostRepositoryInterface { public function all(); public function create(array $data); } Formulating Test Assertions Subsequently, create tests that evaluate this interface through common usage scenarios: public function testAllReturnsCollection() { $mock = Mockery::mock(PostRepositoryInterface::class); $posts = $mock->all(); $this->assertInstanceOf(Collection::class, $posts); } public function testCreateStoresPost() { $mock = Mockery::mock(PostRepositoryInterface::class); $mock->shouldReceive('create')->once(); $mock->create(['title' => 'Test']); } Executing Tests Simply execute phpunit! These contract tests validate our code's public behaviors, independent of implementation specifics. Let's delve into how this approach enhances our code quality over time. Common Areas to Test Contracts Several key areas in a typical Laravel application significantly benefit from contract testing: Database Models & Repositories Classes interacting with the database, such as Eloquent models and repository interfaces, are perfect for contract testing. Verify expected behaviors for retrieving, updating, and associating data without dependence on the backend. API Controllers API surfaces outline your application's public contract with external consumers. Test the adherence of controller methods to formats, expected parameters, and the return of anticipated payloads and status codes. Mailables & Notifications Notifications and mailables facilitate critical communications. Contract test these by ensuring that views and data are rendered correctly, without muddying the tests with transport concerns. Core Application Services Services encapsulate a significant portion of your application's business logic. Test service contracts to ensure the expected payloads and exceptions for a variety of input scenarios, thereby validating core workflows independently of UI logic. By directing tests toward these common artifacts, you can harbor confidence in crucial contracts even as your code evolves. Tests serve as validation and living documentation, delineating intentions and boundaries for key classes. Maintenance and Continuous Testing Contract tests offer ongoing value beyond initial development: Refactoring Without Compromising Contracts Since tests concentrate on public APIs, internal refactors and optimizations won't lead to failures if they don't alter public behavior specifications. This allows secure changes over time. Versioning/Breaking Changes with Clarity Before introducing breaking changes, revise contracts by removing or modifying methods or arguments. This explicitly signals changes to both tests and consumers. Integration with Continuous Integration Incorporate PHPUnit runs into your pipeline. Contract tests now prevent automatic deployment regressions, providing immediate feedback to team members on failures. CI/CD As codebases evolve swiftly, contracts highlight where internal changes alter intended external behaviors. They establish restrictions that maintain backward compatibility over versions. With contracts, releasing confidently becomes an attainable feat. Refactors, features, and optimizations can be implemented safely while preserving established public interfaces. Tests serve as living documentation, reflecting how the code functions and how it's meant to be used. In Conclusion Throughout every development phase, from initial specifications to continual maintenance, ensuring top-notch quality remains paramount. Contract tests provide the structure and assurance that user promises are upheld, even as circumstances evolve. By focusing on expectations instead of implementation, contracts render code understandable from any vantage point. They facilitate safe evolution, ensuring the present operates as before, while also allowing room for innovation. Above all, contract testing transforms relationships. Where fear of breakage once inhibited progress, shared expectations constructed on honesty and respect empower risk-taking. Developers and clients stride together towards a shared destination, bold yet cautious at each step. This alliance of lucidity and care elevates a simple application to the level of a trusted companion. When reliability is guaranteed, so is the freedom to learn, grow, and achieve more together. This represents the true promise of contract testing in Laravel and beyond.

Testing with Laravel Contract-Based Validation

THE TWO GLADIATORS.

Ethereum Is Scaling! Ethereum has seen an explosion in innovation and usage in the last few years. The “Defi summer” of 2020 was the blossoming of decentralized finance, later followed by the success of NFTs reaching the mainstream world. While it was a great party for the Ethereum community, it has also made obvious the scaling limitations of Ethereum. High transaction fees have priced out many users and led them towards alternative Layer 1 blockchains which often make trade-offs with security and decentralization to achieve more scalability. But Ethereum is not giving up on scalability! Ethereum 2.0 will enable a higher transaction capacity, but also, and maybe more importantly, we’ve seen the emergence of Layer 2 scaling, and specifically the idea of a , which is already unfolding before our eyes. rollup-centric roadmap for Ethereum The goal of Layer 2 scaling is to move Ethereum activity off-chain, on second layers. Rollups are a type of Layer 2 solution where the idea is to execute a batch of transactions on the L2 and post them on Ethereum in a compressed form called “rollups.” Obscuro is an L2 rollup solution, alongside Arbitrum, Optimism, StarkWare, Zksync, and others. It means Obscuro users will enjoy fast and cheap transactions. But there’s another dimension to Obscuro that sets it apart from other rollups solution. The Other Battle Scalability has been rightfully identified as the main battle, which Obscuro is proudly fighting alongside his brothers in arms. But there’s a lesser-known battle taking place in the Ethereum world, a darker one, one we don’t talk about as much. Sometimes, we hear frightening rumors and echoes. It is fought in the underworld and the dark forests. The enemy is hidden, invisible, and everywhere, lurking, creeping over us, breathing in our necks, waiting for us to make a mistake. One single mistake, one second of inattention, and it might already be over for us. This is the battle for privacy. And there’s been many casualties. Identity Leaks The more innovations on Ethereum, the more ways you can leak details about yourself and your financial activities. And those details can be used by attackers to target you. NFTs are a good example. It’s hard to resist the temptation of showing off your newly minted NFT or cool ENS domain on social networks, but this effectively links your Ethereum address with your social profile and makes yourself a target of hacking, identity theft, mugging, kidnapping, and blackmailing. For example, . He was one click away from losing $160 million. He linked his social profiles with his ENS domain pointing to an address containing over $160 million in ETH. this seasoned Ethereum user tells a daunting story He was then targeted by a sophisticated social engineering attack which took weeks to unravel and trap him. In the end, he had the lucidity to check the code and escaped, but if we envision billions of people using Ethereum one day, this type of attack will be devastating. In Web3, there is no social safety net to save you from these situations. While there’s a certain level of confidentiality guaranteed by the pseudo-anonymous nature of public blockchains, this is mostly an illusion of privacy, since most on-ramps and off-ramps in and out of Web3 are done via centralized exchanges, which store the real identities of their customers. Web3 services are also a privacy black hole because RPC providers expose users’ IP addresses and other metadata even when no transaction occurs. They could potentially link addresses between themselves and even link them to real identities, and store or share this information. Additionally, non-private transactions are susceptible to aggressive data-mining and harvesting techniques that certain advertising firms use to target non-consenting individuals while abusing their personal privacy. MEV Ethereum transactions are visible to anyone in the mempool before they are executed. It means they can be front-run and value can be extracted from them, a practice known as . The mempool is a scary MEV (Maximal Extractible Value) Dark Forest, and monsters are real. There’s a wide range of MEV attacks, and the MEV monsters are constantly finding new and terrifying ways to extract value from users. If you’re a regular Ethereum user, chances are you’ve already been exploited by MEV, maybe unknowingly. It can be as simple as forgetting to set your slippage on a DEX like Uniswap, and there’s likely a horde of ready to feast on your transaction. sandwich bots By some estimates, front-running was valued at $1.4 billion annually in early 2021. We’ve seen how the public nature of smart contract blockchains like Ethereum makes them a very adversarial environment, not very suited for mainstream adoption. Paradoxically, Web2 is in some ways more private than Web3. While you have to trust a centralized entity in Web2, you generally don’t expose your data to the whole world like you do when you use Ethereum. But friends, hope is not lost! Obscuro: Let There Be Light! Obscuro is combining state-of-the-art scaling knowledge with battle-hardened encryption technology to create an EVM-compatible environment with low gas fees and confidentiality. Not only does Obscuro scale Ethereum, but it also offers encryption thanks to a powerful weapon: the TEE. TEE stands for Trusted Execution Environment. It’s a Secure Enclave found in CPUs where you can execute fully confidential computation. It’s a black box. Obscuro leverages Secure Enclaves to create an optimistic-based roll-up solution on Ethereum where smart contracts are confidential. This is how Obscuro enlightens the dark forests of Ethereum: within Obscuro, your privacy is protected by default, and the mempool is hidden; MEV monsters can’t hurt you there. To our knowledge, there is only one other brave warrior attempting to fight for privacy with an L2 roll-up solution on Ethereum; it’s Aztec and they are using zero-knowledge technology. Obscuro developer Tudor Malene explains in why he thinks Secure Enclaves is a better-suited weapon to achieve general-purpose privacy. this post This battle for privacy is not unique to Ethereum, and Obscuro can count on some solid allies like Secret Network on Cosmos and Phala on Polkadot. Together they are the three muskeTEErs of privacy, united by the TEEs they use to slay their enemies. A New World We’ve seen how Obscuro keeps you safe from the lurking threats of the Ethereum world, but privacy is not just about keeping you safe and protecting your identity, it’s also about enabling new blockchain use cases and inviting new participants to take advantage of Web3. Financial markets present a compelling environment where privacy is needed to prevent inadvertent disclosure of positions and front-running of trades. In traditional markets, , and are commonly used to exchange assets in a private manner. dark pools OTC trading, sealed auctions Obscuro is an ideal platform for those use cases to bloom and onboard a new class of institutions and individuals to Ethereum. Obscuro is also a welcoming place to empower businesses because they rely on privacy to prevent competitors from seeing valuable proprietary information like supply chain partners, investment strategies, primary sources of profit, etc. And maybe it’s in the where the full potential of confidential computation is unlocked, who knows? As a general-purpose privacy solution, Obscuro allows entire smart contracts to be hidden. metaverse And there are a lot of exciting ways to experiment with NFTs and gaming when you can hide the code. With Obscuro, we could make a game where players earn loot boxes NFTs and can exchange them on a market, even if they’re unopened. We need the private computation of Obscuro to prevent people from looking at the NFT metadata and seeing exactly what’s inside the boxes. Loot boxes and nested NFTs: We can build on-chain versions of classic games like or where it’s essential your opponent doesn’t know your strategy or your cards. Poker Battleship As an artist, you can hide exclusive content in your NFTs, which only the owner of the NFT is able to see. Exclusive content: : We can deploy the entire plot of an adventure game or a treasure quest on-chain, and no one will be able to look up the storyline in the code to cheat! Game lores : We could imagine an NFT collection where attributes evolve based on any blockchain or real-world event, and no one could predict the changes because the code is private. Let’s say you’ve minted a character that looks basic at first, but you realize he morphs into a wolf every full moon. Wow, you’ve actually minted a super rare werewolf! Shapeshifting NFTs And certainly, a myriad of other use cases we haven’t even thought about. Obscuro welcomes new and creative ways to leverage scalability and privacy. If you have an idea, let us know! Find Out More If you’re interested in learning more about , check out our or dive into our . Chat with the community on and follow us on . Obscuro litepaper whitepaper Discord Twitter Also published here

Introducing Obscuro - An Encrypted Layer 2 on Ethereum

Presentamos Obscuro: una capa 2 cifrada en Ethereum

Présentation d'Obscuro - Une couche 2 cryptée sur Ethereum

ऑब्स्कुरो का परिचय - एथेरियम पर एक एन्क्रिप्टेड परत 2

Obscuro の紹介 - イーサリアム上の暗号化レイヤー 2

Apresentando Obscuro - Uma Camada Criptografada 2 no Ethereum

Giới thiệu Obscuro - Lớp 2 được mã hóa trên Ethereum

Obscuro 简介 - 以太坊上的加密第 2 层

The Murder on the Links by Agatha Christie, is part of the HackerNoon Books Series. You can jump to any chapter in this book here . We Make Further Investigations XVII. We Make Further Investigations I have set down the Beroldy case in full. Of course all the details did not present themselves to my memory as I have recounted them here. Nevertheless, I recalled the case fairly accurately. It had attracted a great deal of interest at the time, and had been fully reported by the English papers, so that it did not need much effort of memory on my part to recollect the salient details. Just for the moment, in my excitement, it seemed to clear up the whole matter. I admit that I am impulsive, and Poirot deplores my custom of jumping to conclusions, but I think I had some excuse in this instance. The remarkable way in which this discovery justified Poirot’s point of view struck me at once. “Poirot,” I said, “I congratulate you. I see everything now.” “If that is indeed the truth, I congratulate you, mon ami. For as a rule you are not famous for seeing—eh, is it not so?” I felt a little annoyed. “Come now, don’t rub it in. You’ve been so confoundedly mysterious all along with your hints and your insignificant details that any one might fail to see what you were driving at.” Poirot lit one of his little cigarettes with his usual precision. Then he looked up. “And since you see everything now, mon ami, what exactly is it that you see?” “Why, that it was Madame Daubreuil—Beroldy, who murdered Mr. Renauld. The similarity of the two cases proves that beyond a doubt.” “Then you consider that Madame Beroldy was wrongly acquitted? That in actual fact she was guilty of connivance in her husband’s murder?” I opened my eyes wide. “But of course! Don’t you?” Poirot walked to the end of the room, absentmindedly straightened a chair, and then said thoughtfully. “Yes, that is my opinion. But there is no ‘of course’ about it, my friend. Technically speaking, Madame Beroldy is innocent.” “Of that crime, perhaps. But not of this.” Poirot sat down again, and regarded me, his thoughtful air more marked than ever. “So it is definitely your opinion, Hastings, that Madame Daubreuil murdered M. Renauld?” “Yes.” “Why?” He shot the question at me with such suddenness that I was taken aback. “Why?” I stammered. “Why? Oh, because—” I came to a stop. Poirot nodded his head at me. “You see, you come to a stumbling-block at once. Why should Madame Daubreuil (I shall call her that for clearness sake) murder M. Renauld? We can find no shadow of a motive. She does not benefit by his death; considered as either mistress or blackmailer she stands to lose. You cannot have a murder without a motive. The first crime was different, there we had a rich lover waiting to step into her husband’s shoes.” “Money is not the only motive for murder,” I objected. “True,” agreed Poirot placidly. “There are two others, the crime passionnel is one. And there is the third rare motive, murder for an idea which implies some form of mental derangement on the part of the murderer. Homicidal mania, and religious fanaticism belong to that class. We can rule it out here.” “But what about the crime passionnel? Can you rule that out? If Madame Daubreuil was Renauld’s mistress, if she found that his affection was cooling, or if her jealousy was aroused in any way, might she not have struck him down in a moment of anger?” Poirot shook his head. “If—I say if, you note—Madame Daubreuil was Renauld’s mistress, he had not had time to tire of her. And in any case you mistake her character. She is a woman who can simulate great emotional stress. She is a magnificent actress. But, looked at dispassionately, her life disproves her appearance. Throughout, if we examine it, she had been cold-blooded and calculating in her motives and actions. It was not to link her life with that of her young lover that she connived at her husband’s murder. The rich American, for whom she probably did not care a button, was her objective. If she committed a crime, she would always do so for gain. Here there was no gain. Besides, how do you account for the digging of the grave? That was a man’s work.” “She might have had an accomplice,” I suggested, unwilling to relinquish my belief. “I pass to another objection. You have spoken of the similarity between the two crimes. Wherein does that lie, my friend?” I stared at him in astonishment. “Why, Poirot, it was you who remarked on that! The story of the masked men, the ‘secret,’ the papers!” Poirot smiled a little. “Do not be so indignant, I beg of you. I repudiate nothing. The similarity of the two stories links the two cases together inevitably. But reflect now on something very curious. It is not Madame Daubreuil who tells us this tale—if it were all would indeed be plain sailing—it is Madame Renauld. Is she then in league with the other?” “I can’t believe that,” I said slowly. “If it is so, she must be the most consummate actress the world has ever known.” “Ta-ta-ta,” said Poirot impatiently. “Again you have the sentiment, and not the logic! If it is necessary for a criminal to be a consummate actress, then by all means assume her to be one. But is it necessary? I do not believe Madame Renauld to be in league with Madame Daubreuil for several reasons, some of which I have already enumerated to you. The others are self-evident. Therefore, that possibility eliminated, we draw very near to the truth which is, as always, very curious and interesting.” “Poirot,” I cried, “what more do you know?” “Mon ami, you must make your own deductions. You have ‘access to the facts!’ Concentrate your grey cells. Reason—not like Giraud—but like Hercule Poirot.” “But are you sure?” “My friend, in many ways I have been an imbecile. But at last I see clearly.” “You know everything?” “I have discovered what M. Renauld sent for me to discover.” “And you know the murderer?” “I know one murderer.” “What do you mean?” “We talk a little at cross-purposes. There are here not one crime, but two. The first I have solved, the second—eh bien, I will confess, I am not sure!” “But, Poirot, I thought you said the man in the shed had died a natural death?” “Ta-ta-ta.” Poirot made his favourite ejaculation of impatience. “Still you do not understand. One may have a crime without a murderer, but for two crimes it is essential to have two bodies.” His remark struck me as so peculiarly lacking in lucidity that I looked at him in some anxiety. But he appeared perfectly normal. Suddenly he rose and strolled to the window. “Here he is,” he observed. “Who?” “M. Jack Renauld. I sent a note up to the Villa to ask him to come here.” That changed the course of my ideas, and I asked Poirot if he knew that Jack Renauld had been in Merlinville on the night of the crime. I had hoped to catch my astute little friend napping, but as usual, he was omniscient. He, too, had inquired at the station. “And without doubt we are not original in the idea, Hastings. The excellent Giraud, he also has probably made his inquiries.” “You don’t think—” I said, and then stopped. “Ah, no, it would be too horrible!” Poirot looked inquiringly at me, but I said no more. It had just occurred to me that though there were seven women directly or indirectly connected with the case Mrs. Renauld, Madame Daubreuil and her daughter, the mysterious visitor, and the three servants—there was, with the exception of old Auguste who could hardly count, only one man—Jack Renauld. And a man must have dug a grave. … I had no time to develop further the appalling idea that had occurred to me, for Jack Renauld was ushered into the room. Poirot greeted him in a business-like manner. “Take a seat, monsieur. I regret infinitely to derange you, but you will perhaps understand that the atmosphere of the Villa is not too congenial to me. M. Giraud and I do not see eye to eye about everything. His politeness to me has not been striking and you will comprehend that I do not intend any little discoveries I may make to benefit him in any way.” “Exactly, M. Poirot,” said the lad. “That fellow Giraud is an ill-conditioned brute, and I’d be delighted to see some one score at his expense.” “Then I may ask a little favour of you?” “Certainly.” “I will ask you to go to the railway station and take a train to the next station along the line, Abbalac. Ask there at the cloak-room whether two foreigners deposited a valise there on the night of the murder. It is a small station, and they are almost certain to remember. Will you do this?” “Of course I will,” said the boy, mystified, though ready for the task. “I and my friend, you comprehend, have business elsewhere,” explained Poirot. “There is a train in a quarter of an hour, and I will ask you not to return to the Villa, as I have no wish for Giraud to get an inkling of your errand.” “Very well, I will go straight to the station.” He rose to his feet. Poirot’s voice stopped him. “One moment, M. Renauld, there is one little matter that puzzles me. Why did you not mention to M. Hautet this morning that you were in Merlinville on the night of the crime?” Jack Renauld’s face went crimson. With an effort he controlled himself. “You have made a mistake. I was in Cherbourg, as I told the examining magistrate this morning.” Poirot looked at him, his eyes narrowed, cat-like, until they only showed a gleam of green. “Then it is a singular mistake that I have made there—for it is shared by the station staff. They say you arrived by the 11:40 train.” For a moment Jack Renauld hesitated, then he made up his mind. “And if I did? I suppose you do not mean to accuse me of participating in my father’s murder?” He asked the question haughtily, his head thrown back. “I should like an explanation of the reason that brought you here.” “That is simple enough. I came to see my fiancée, Mademoiselle Daubreuil. I was on the eve of a long voyage, uncertain as to when I should return. I wished to see her before I went, to assure her of my unchanging devotion.” “And you did see her?” Poirot’s eyes never left the other’s face. There was an appreciable pause before Renauld replied. Then he said: “Yes.” “And afterwards?” “I found I had missed the last train. I walked to St. Beauvais where I knocked up a garage and got a car to take me back to Cherbourg.” “St. Beauvais? That is fifteen kilometres. A long walk, M. Renauld.” “I—I felt like walking.” Poirot bowed his head as a sign that he accepted the explanation. Jack Renauld took up his hat and cane and departed. In a trice Poirot jumped to his feet. “Quick, Hastings. We will go after him.” Keeping a discreet distance behind our quarry, we followed him through the streets of Merlinville. But when Poirot saw that he took the turning to the station, he checked himself. “All is well. He has taken the bait. He will go to Abbalac, and will inquire for the mythical valise left by the mythical foreigners. Yes, mon ami, all that was a little invention of my own.” “You wanted him out of the way!” I exclaimed. “Your penetration is amazing, Hastings! Now, if you please, we will go up to the Villa Geneviève.” 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. Agatha Christie (2019). The Murder on the Links. Urbana, Illinois: Project Gutenberg. Retrieved https://www.gutenberg.org/cache/epub/58866/pg58866-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 www.gutenberg.org , located at https://www.gutenberg.org/policy/license.html.

We Make Further Investigations

A Special Providence

Theoretical writings on Architecture

The A B C of Relativity by Bertrand Russells, is part of the HackerNoon Books Series. You can jump to any chapter in this book here . XIII. THE ABOLITION OF “FORCE” XIII. THE ABOLITION OF “FORCE” In the Newtonian system, bodies under the action of no forces move in straight lines with uniform velocity; when bodies do not move in this way, their change of motion is ascribed to a “force.” Some forces seem intelligible to our imagination: those exerted by a rope or string, by bodies colliding, or by any kind of obvious pushing or pulling. As explained in an earlier chapter, our apparent imaginative understanding of these processes is quite fallacious; all that it really means is that past experience enables us to foresee more or less what is going to happen without the need of mathematical calculations. But the “forces” involved in gravitation and in the less familiar forms of electrical action do not seem in this way “natural” to our imagination. It seems odd that the earth can float in the void: the natural thing to suppose is that it must fall. That is why it has to be supported on an elephant, and the [Pg 193]elephant on a tortoise, according to some early speculators. The Newtonian theory, in addition to action at a distance, introduced two other imaginative novelties. The first was, that gravitation is not always and essentially directed what we should call “downwards,” i.e., towards the center of the earth. The second was, that a body going round and round in a circle with uniform velocity is not “moving uniformly” in the sense in which that phrase is applied to the motion of bodies under no forces, but is perpetually being turned out of the straight course towards the center of the circle, which requires a force pulling it in that direction. Hence Newton arrived at the view that the planets are attracted to the sun by a force, which is called gravitation. This whole point of view, as we have seen, is superseded by relativity. There are no longer such things as “straight lines” in the old geometrical sense. There are “straightest lines,” or geodesics, but these involve time as well as space. A light ray passing through the solar system does not describe the same orbit as a comet, from a geometrical point of view; nevertheless each moves in a geodesic. The whole imaginative picture is changed. A poet might say that water [Pg 194]runs down hill because it is attracted to the sea, but a physicist or an ordinary mortal would say that it moves as it does, at each point, because of the nature of the ground at that point, without regard to what lies ahead of it. Just as the sea does not cause the water to run towards it, so the sun does not cause the planets to move round it. The planets move round the sun because that is the easiest thing to do—in the technical sense of “least action.” It is the easiest thing to do because of the nature of the region in which they are, not because of an influence emanating from the sun. The supposed necessity of attributing gravitation to a “force” attracting the planets towards the sun has arisen from the determination to preserve Euclidean geometry at all costs. If we suppose that our space is Euclidean, when in fact it is not, we shall have to call in physics to rectify the errors of our geometry. We shall find bodies not moving in what we insist upon regarding as straight lines, and we shall demand a cause for this behavior. Eddington has stated this matter with admirable lucidity. He supposes a physicist who has assumed the formula for interval which is used in the special [Pg 195]theory of relativity—a formula which still supposes that the observer’s space is Euclidean. He continues: Since intervals can be compared by experimental methods, he ought soon to discover that his (formula for the interval) cannot be reconciled with observational results, and so realize his mistake. But the mind does not so readily get rid of an obsession. It is more likely that our observer will continue in his opinion, and attribute the discrepancy of the observations to some influence which is present and affects the behavior of his test-bodies. He will, so to speak, introduce a supernatural agency which he can blame for the consequences of his mistake.... The name given to any agency which causes deviation from uniform motion in a straight line is force according to the Newtonian definition of force. Hence the agency invoked through our observer’s mistake is described as a “field of force.”... A field of force represents the discrepancy between the naturalgeometry of a co-ordinate system and the abstractgeometry arbitrarily ascribed to it. [15] If people were to learn to conceive the world in the new way, without the old notion of “force,” it would alter not only their physical imagination, but probably also their morals and politics. The latter [Pg 196]effect would be quite illogical, but is none the less probable on that account. In Newton’s theory of the solar system, the sun seems like a monarch whose behests the planets have to obey. In Einstein’s world there is more individualism and less government than in Newton’s. There is also far less hustle: we have seen that laziness is the fundamental law of Einstein’s universe. The word “dynamic” has come to mean, in newspaper language, “energetic and forceful”; but if it meant “illustrating the principles of dynamics,” it ought to be applied to the people in hot climates who sit under banana trees waiting for the fruit to drop into their mouths. I hope that journalists, in future, when they speak of a “dynamic personality,” will mean a person who does what is least trouble at the moment, without thinking of remote consequences. If I can contribute to this result, I shall not have written in vain. It has been customary for people to draw arguments from the laws of nature as to what we ought to do. Such arguments seem to me a mistake: to imitate nature may be merely slavish. But if nature, as portrayed by Einstein, is to be our model, it would seem that the anarchists will [Pg 197]have the best of the argument. The physical universe is orderly, not because there is a central government, but because every body minds its own business. No two particles of matter ever come into contact; when they get too close, they both move off. If a man were had up for knocking another man down, he would be scientifically correct in pleading that he had never touched him. What happened was that there was a hill in space-time in the region of the other man’s nose, and it fell down the hill. The abolition of “force” seems to be connected with the substitution of sight for touch as the source of physical ideas, as explained in . When an image in a looking glass moves, we do not think that something has pushed it. In places where there are two large mirrors opposite to each other, you may see innumerable reflections of the same object. Suppose a gentleman in a top-hat is standing between the mirrors, there may be twenty or thirty top-hats in the reflections. Suppose now somebody comes and knocks off the gentleman’s hat with a stick: all the other twenty or thirty top-hats will tumble down at the same moment. We think that a force is needed to knock off the “real” [Pg 198]top-hat, but we think the remaining twenty or thirty tumble off, so to speak, of themselves, or out of a mere passion for imitation. Let us try to think out this matter a little more seriously. Chapter I Obviously something happens when an image in a looking glass moves. From the point of view of sight, the event seems just as real as if it were not in a mirror. But nothing has happened from the point of view of touch or hearing. When the “real” top-hat falls, it makes a noise; the twenty or thirty reflections fall without a sound. If it falls on your toe, you feel it; but we believe that the twenty or thirty people in the mirrors feel nothing, though top-hats fall on their toes too. But all this is equally true of the astronomical world. It makes no noise, because sound cannot travel across a vacuum. So far as we know, it causes no “feelings,” because there is no one on the spot to “feel” it. The astronomical world, therefore, seems hardly more “real” or “solid” than the world in the looking glass, and has just as little need of “force” to make it move. The reader may feel that I am indulging in idle sophistry. “After all,” he may say, “the image in the mirror is the reflection of something [Pg 199]solid, and the top-hat in the mirror only falls off because of the force applied to the real top-hat. The top-hat in the mirror cannot indulge in behavior of its own; it has to copy the real one. This shows how different the image is from the sun and the planets, because they are not obliged to be perpetually imitating a prototype. So you had better give up pretending that an image is just as real as one of the heavenly bodies.” There is, of course, some truth in this; the point is to discover exactly what truth. In the first place, images are not “imaginary.” When you see an image, certain perfectly real light waves reach your eye; and if you hang a cloth over the mirror, these light waves cease to exist. There is, however, a purely optical difference between an “image” and a “real” thing. The optical difference is bound up with this question of imitation. When you hang a cloth over the mirror, it makes no difference to the “real” object; but when you move the “real” object away, the image vanishes also. This makes us say that the light rays which make the image are only reflected at the surface of the mirror, and do not really come from a point behind it, but from [Pg 200]the “real” object. We have here an example of a general principle of great importance. Most of the events in the world are not isolated occurrences, but members of groups of more or less similar events, which are such that each group is connected in an assignable manner with a certain small region of space-time. This is the case with the light rays which make us see both the object and its reflection in the mirror: they all emanate from the object as a center. If you put an opaque globe round the object at a certain distance, the object and its reflection are invisible at any point outside the globe. We have seen that gravitation, although no longer regarded as an action at a distance, is still connected with a center: there is, so to speak, a hill symmetrically arranged about its summit, and the summit is the place where we conceive the body to be which is connected with the gravitational field we are considering. For simplicity, common sense lumps together all the events which form one group in the above sense. When two people see the same object, two different events occur, but they are events belonging to one group and connected with the same center. Just the same applies when two people (as we say) hear the [Pg 201]same noise. And so the reflection in a mirror is less “real” than the object reflected, even from an optical point of view, because light rays do not spread in all directions from the place where the image seems to be, but only in directions in front of the mirror, and only so long as the object reflected remains in position. This illustrates the usefulness of grouping connected events about a center in the way we have been considering. When we examine the changes in such a group of objects, we find that they are of two kinds: there are those which affect only some member of the group, and those which make connected alterations in all the members of the group. If you put a candle in front of a mirror, and then hang black cloth over the mirror, you alter only the reflection of the candle as seen from various places. If you shut your eyes, you alter its appearance to you, but not its appearance elsewhere. If you put a red globe round it at a distance of a foot, you alter its appearance at any distance greater than a foot, but not at any distance less than a foot. In all these cases, you do not regard the candle itself as having changed; in fact, in all of them, you find that [Pg 202]there are groups of changes connected with a different center or with a number of different centers. When you shut your eyes, for instance, your eyes, not the candle, look different to any other observer: the center of the changes that occur is in your eyes. But when you blow out the candle, its appearance everywhere is changed; in this case you say that the change has happened to the candle. The changes that happen to an object are those that affect the whole group of events which center about the object. All this is only an interpretation of common sense, and an attempt to explain what we mean by saying that the image of the candle in the mirror is less “real” than the candle. There is no connected group of events situated all round the place where the image seems to be, and changes in the image center about the candle, not about a point behind the mirror. This gives a perfectly verifiable meaning to the statement that the image is “only” a reflection. And at the same time it enables us to regard the heavenly bodies, although we can only see and not touch them, as more “real” than an image in a looking glass. We can now begin to interpret the common sense notion of one body [Pg 203]having an “effect” upon another, which we must do if we are really to understand what is meant by the abolition of “force.” Suppose you come into a dark room and switch on the electric light: the appearance of everything in the room is changed. Since everything in the room is visible because it reflects the electric light, this case is really analogous to that of the image in the mirror; the electric light is the center from which all the changes emanate. In this case, the “effect” is explained by what we have already said. The more important case is when the effect is a movement. Suppose you let loose a tiger in the middle of a Bank Holiday crowd: they would all move, and the tiger would be the center of their various movements. A person who could see the people but not the tiger would infer that there was something repulsive at that point. We say in this case that the tiger has an effect upon the people, and we might describe the tiger’s action upon them as of the nature of a repulsive force. We know, however, that they fly because of something which happens to them, not merely because the tiger is where he is. They fly because they can see and hear him, that is to say, because certain waves reach their eyes and [Pg 204]ears. If these waves could be made to reach them without there being any tiger, they would fly just as fast, because the neighborhood would seem to them just as unpleasant. Let us now apply similar considerations to the sun’s gravitation. The “force” exerted by the sun only differs from that exerted by the tiger in being attractive instead of repulsive. Instead of acting through waves of light or sound, the sun acquires its apparent power through the fact that there are modifications of space-time all round the sun. Like the noise of the tiger, they are more intense near their source; as we travel away they grow less and less. To say that the sun “causes” these modifications of space-time is to add nothing to our knowledge. What we know is that the modifications proceed according to a certain rule, and that they are grouped symmetrically about the sun as center. The language of cause and effect adds only a number of quite irrelevant imaginings, connected with will, muscular tension, and such matters. What we can more or less ascertain is merely the formula according to which space-time is modified by the presence of gravitating matter. [Pg 205]More correctly: we can ascertain what kind of space-time is the presence of gravitating matter. When space-time is not accurately Euclidean in a certain region, but has a non-Euclidean character which grows more and more marked as we approach a certain center, and when, further, the departure from Euclid obeys a certain law, we describe this state of affairs briefly by saying that there is gravitating matter at the center. But this is only a compendious account of what we know. What we know is about the places where the gravitating matter is not, not about the place where it is. The language of cause and effect (of which “force” is a particular case) is thus merely a convenient shorthand for certain purposes; it does not represent anything that is genuinely to be found in the physical world. And how about matter? Is matter also no more than a convenient shorthand? This question, however, being a large one, demands a separate chapter. 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. Bertrand Williams (2004). THE A B C OF RELATIVITY. Urbana, Illinois: Project Gutenberg. Retrieved October 2022, from https://www.gutenberg.org/files/67104/67104-h/67104-h.htm 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 www.gutenberg.org , located at https://www.gutenberg.org/policy/license.html .

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