EQUITY & PROPERTY LAW Essay Example | Topics and Well Written Essays - 1000 words

EQUITY & PROPERTY LAW - Essay Example When equity developed as a parallel system to the common law, it was considered innovative by acknowledging â€Å"new† rights where common law failed to provide â€Å"justice†2. The intrinsic nature of this innovative system lay in the judicial â€Å"discretion† referred to by Lord Hoffman in Co-operative Insurance Society Limited v Argyll Stores Holdings Limited3. From a historical perspective, equity developed as a result of inflexibility of common law and â€Å"wiped away the tears of the common law†4. However, this exercise of judicial discretion led to an uneasy relationship with the common law. The clash was resolved in favour of equity and resulted in equity prevailing in the event of a conflict, which is now statutorily enshrined in section 49 of the Supreme Court Act 1981. Moreover, if we firstly consider Bill and Muriel’s position, if Charles had orally agreed for them to acquire proprietary interests in the Property, then Section 2 of the Law of Property (Miscellaneous Provisions) Act 1989 (the 1989 Act) provides that â€Å"a contract for the sale or other disposition of an interest in land can only be made in writing and only by incorporating all the terms, which the parties have expressly agreed in one document or, where the contracts have been exchanged, in each†. The essence of Section 2 is the requirement that the contract must be in writing and contain all the terms expressly agreed to and be signed by both parties. If the rules are not complied with, there will be no contract. In the past, failure to comply with the written requirements was remedied by equity when there had been part performance of a contract. Whilst there is no express provision in the 1989 Act specifically abolishing part performance, there has been an assumption that the doctrine is no longer applicable as section 2 clearly renders oral contracts void. Therefore, at common law, any oral

Assessment case study Essay Example | Topics and Well Written Essays - 2000 words

Assessment case study - Essay Example The two most important models related to disability are the medical and social models. For assessment of John's case, I would follow both the models as "Both the medical and social model aim to 'cure' disability, the former by curing the impairment and the latter by curing the environment and social attitudes. This indicates that, despite the polarity of their views of the world and the opposing positions they occupy, both models aim to improve the state of disability." (Wilson, 2003. p. 22) When a disabled child is small, his condition does not bother his parents much, as bringing him up is like bringing up a normal child, as at that stage all children are dependent on their parents for their needs. It is when the child grows up (during their adolescence stage) as is the case of John who is 15, that the parents as well as the disabled child realize the reality that that their situation is different from the rest and is going to remain the same. This is the time when parents get tota lly dejected. Since I am going to meet John and his family at such a critical stage, I would keep in mind that he and his family need emotional support more than anything else. For family members the burden of shouldering the responsibility of a disabled person takes a toll on their psyche and might frustrate them so much, that they start abusing and cursing the disabled person. They might not allow the disabled person to do things according to his liking by pointing his incapability to him. During the assessment meeting I would look into this aspect to determine whether John is getting proper care and love by his parents and siblings or not. If I find something amiss I will try to focus on this aspect so that John and his family members are able to deal with their unfortunate situation in a better way. I would see to it that every positive change begins from home. Since John is so used to of his family members being an intermediary between him and outsiders, I will try to encourage John to express himself as much as possible in the first meeting so that he gets comfortable with me and sheds his hesitance as my main objective is to improve his condition more than that of his family members. Both John and I will have to struggle a lot to understand each other because of his critical condition. I will try to systematically organize the service delivery in accordance to the policies and programmes designed by my team to deliver the best possible service to John and his family. Since assessment meeting is not only about John, but his circumstances too, I will try my best to organise, systemise and rationalise the information provided by John's father in his letter as well as my face-to-face communication with his entire family sensitively to get the crux of the whole situation. In this case, I will totally abide by the viewpoint of Coulshed and Orme who have rightly pointed out that "assessment is not just an event, for example the production of a profile on someone or a report for the court; it is, as indicated, a way of continuously collecting and synthesising available data, which includes thoughts and feelings, in order to formulate 'treatment' plans." (1998, p. 21) Through the letter of John's father it as clear that in the past he has not got much help from the

Reaction against hydrogen peroxide Essay Example for Free

Reaction against hydrogen peroxide Essay The action of catalase on hydrogen peroxide Aim The aim of this experiment is to discover the relationship between the concentration of the substrate and the rate of the reaction catalysed by an enzyme, by looking at the decomposition of hydrogen peroxide under the action of catalase, and to determine a value for Vmax and the Michaelis constant for catalase. Background Theory An enzyme is a protein biological catalyst. Catalysts speed up or slow down the rate at which chemical reactions occur. They are not used up in the reactions and can be retrieved unchanged afterwards. Biological catalysts control the rate of reactions in living things. Each enzyme is substrate specific it can control only one reaction. For example, the digestion of starch is begun in the mouth by the enzyme amylase. An equation for this reaction can be shown like this: Amylase Starch Simple sugars The enzyme only facilitates the reaction, it is not used up. Each molecule of enzyme can be reused indefinitely, unless it is damaged, or denatured. Enzymes are proteins, so they are denatured if the polypeptide chains, which are precisely coiled and folded to form the active site, become unfolded by the kinetic energy from heat, or the covalent bonds are disrupted. Whilst some heat will increase the rate of reaction because of the increased number of collisions between enzyme and substrate, too much heat will denature the enzyme and render it completely ineffective. Enzymes are also affected by the pH at which they have to work. Charged hydrogen or hydroxide ions in acids or alkalis can cancel out the charges on the active sites of the enzymes, and render them ineffective. Hydrogen peroxide is a toxin produced in every cell of living organisms as a by-product of respiration. It is the same chemical that is used to bleach hair, and so must be broken down before it can damage the cells. It decomposes to give water and oxygen. This reaction will occur naturally, but at a very slow rate. To speed it up an enzyme is used. The enzyme which catalyses the decomposition of hydrogen peroxide is called catalase. Catalase is found in all living cells to decompose the hydrogen peroxide. In this experiment it is being obtained from live yeast in a suspension. The method by which catalase works is called the lock and key method. Catalase works because it has an active site. At this point the enzyme attaches to the hydrogen peroxide molecule, because the opposite charges of enzyme and substrate attract each other, forming an enzyme substrate complex. The enzyme catalyses the reaction, and then the new charges on the product repel the enzyme away to act on a new substrate molecule. (see fig 1) The decomposition of hydrogen peroxide has the following formula: catalase 2H2O2 2H2O + O2 The rate of a reaction is a measure of the change in the amount of reactant or product with time. The rate of decomposition H2O2 can be measured using the volume of oxygen produced, from the formula: Rate of reaction = change in amount of product time The rate of reaction is determined by collision theory:- For a reaction between two substances to occur, the enzyme and substrate particles must collide with each other. If more collisions occur in a reaction, rate will increase. If the reactant particles gain energy and collide faster, then each collision will have more energy, there will be more successful collisions, and rate will also increase. In this case, the more collisions between hydrogen peroxide and catalase molecules, the more hydrogen peroxide will decompose. Anything which increases collisions will increase rate. Increasing the concentration of the substrate (hydrogen peroxide) solution means that there are more substrate molecules in the same volume, causing more collisions, and thus increasing rate. The rate of reaction changes with concentration, but the overall yield of oxygen is independent of factors affecting the rate, so measuring the amount of oxygen produced over the whole reaction is meaningless. Instead, the initial rate of reaction can be estimated by measuring the volume of oxygen produced in the early stages of the reaction. This value can then be compared between the different concentrations of hydrogen peroxide, and used to plot a graph of substrate concentration against rate, from which values for Vmax and the Michaelis constant (Km) can be obtained. The relationship between substrate concentration and rate of reaction is described by the Michaelis-Menton equation: v = Vmax [S] Km + [S]. Vmax is a measure of the maximum rate at which an enzyme can act, and it is the horizontal asymptote of the graph of substrate against time that is when the amount of enzyme is the limiting factor. The Km is defined as the substrate concentration at which the rate of enzyme action is half Vmax. It is measure of the affinity of an enzyme for its substrate molecule the higher Km, the weaker the binding force between the enzyme and substrate. Both Km and Vmax are constants at a specific enzyme concentration and temperature. Pilot Experiment A pilot experiment was carried out in order to see whether the method was practical and could produce good, reliable results, and to choose the concentrations of hydrogen peroxide and the length of time over which the oxygen would be collected so that no more than 50cm3 of gas was given off. Pilot Method 1. 10cm3 of hydrogen peroxide solution was measured into boiling tubes using a syringe, and the apparatus set up as shown below. 2. Using a 1cm3 syringe the yeast suspension was added to the boiling tube and the stopwatch started. 3. Thirty seconds was timed, and then the burette was taken off the end of the delivery tube, but not out of the water, and the volume of gas collected was measured. 4. This was recorded and repeated for each concentration of hydrogen peroxide, made up as shown in the dilution tables below. Pilot Results Concentration of hydrogen peroxide Initial reading on burette (cm3) Final reading on burette (cm3) Volume of gas collected (cm3) Rate of reaction (cm3/s) to 2dOff scale. Analysis of and Modifications to the Pilot These results show a clear increase in the volume of gas collected as the concentration increases, suggesting that with modifications this method will enable clear conclusions to be drawn. Carrying the experiment out over 30s resulted in too much gas being produced at the highest concentration to be recorded with the apparatus available, so for the main experiment the oxygen will be collected over 15s. More readings will be taken to enable a more reliable graph to be drawn at9 and 20%. The experiment will be repeated three times and any anomalous results will be identified and excluded from the average in order to enable more reliable results. Prediction I predict that initially the rate of reaction will increase with the concentration. As the concentration of hydrogen peroxide increases so will the number of collisions between enzyme and substrate molecules, so the hydrogen peroxide will decompose faster into water and oxygen. I predict that this reaction will obey Michaelis-Menton kinetics, and that the graph of rate of reaction against hydrogen peroxide concentration will give a rectangular hyperbola as shown below: The increase in rate of reaction will not continue indefinitely there will be an asymptote when Rate = Vmax, when all the catalase molecules are catalysing the reaction as fast as possible, and so the rate cannot increase without supplying more enzyme. Main Experiment Plan Fair Test A fair test is one from which a reliable conclusion can be drawn. For a fair test only one variable must be changed at a time. In this experiment the concentration of hydrogen peroxide is being changed, and so all others must be controlled. Variables. Independent Variable:- concentration of hydrogen peroxide Dependent Variable:- volume of oxygen gas collected in 15s Controlled Variables:- temperature, volume of hydrogen peroxide, amount of yeast, apparatus, time.   The reaction will be carried out in a water bath at 20? C. Since water is a good thermal buffer it should be fairly easy to keep the temperature constant.   Volume of hydrogen peroxide solution will be controlled quite easily by using two syringes to measure the water and hydrogen peroxide volumes as dictated by the dilution table below.

Colgate Marketing Plan analysis

Colgate Marketing Plan analysis Colgate has realised the significance of marketing plan in its 200 years of existence. This is essential towards understanding the business plans, aims, and communicating the companys horizon within the operational market. This report will focus on the marketing plan for the new Colgate toothbrush called 360 Actiflex, as part of Oral care and Health products which comprises of Tooth paste, Tooth brush, Mouth wash and dental floss. Oral care is important because research indicate that a clean mouth prevents gum disease and prevent heart disease. The plan will address the niche market of dental professional which will recommend the product to their clients and the product brand name will remain to be Colgate. Situation Analysis Colgate Palmolive has been in existence for over 200 hundred years and they operate in more than 70 countries such as Brazil, America, Latin America, Asian etc which will enable to penetrate the new market easily. This is as a result of their brand, experience, capacity in term of resources and existing structure. Product Classification The 360 Actiflex brush with flexible head for cheek and tongue cleaner is design to clean all direction, to remove bacteria from your teeth, cheek, gum and tongue for healthier mouth. Customers want toothbrush that can give them healthy clean teeth and breathe at a reasonable price. Market Summary Colgate Palmolive is aiming to increase its market share in toothbrush which is currently at 30% in 2009 by demonstrating proficiency and professionalism by serving new and existing customers. Through the RD, Colgate has been able to improve their existing products and introduce more products which meet the customers need in dental care. Market Needs According to research, customers only need two or three toothbrushes a year. 65% of customers have more than one toothbrush, 24% keep brush at work 54% have special toothbrush for travelling. Buying Behaviour Pattern Customers base their purchase of tooth brush on how comfortable it feels in their mouth, whether it is soft enough and suits their mouth size or flexible enough. Their purchase is also base on the ability of the toothbrush to remove food particles, plaque removal and gum stimulation. SWOT ANALYSIS Strengths Colgate maintains strong relationship with dental professional, wholesalers and retailers. Have a good communication with the consumers through media and other social resources Extensive industry knowledge and expertise Well designed training programme. Ability to convince customers to spend more though continual rebranding and growing company sales by releasing new twists on under established brands. Weaknesses Increase competition where other competitors produce the same products. The recent recession has impact on the reduction of planned expansion. Opportunities Oral care is a growing industry with the ever changing customers taste. Continuous innovation using the idea technology challenge. Threats Changes in exchange rates for foreign currencies, which may reduce the U.S. dollar value of revenue, receive from oversea market sales. Political or economic instability or changing macroeconomic conditions in our major markets, Lack of well-established or reliable legal systems in certain areas where the Company operates, Objective Setting Colgate by introducing the 360 Actiflex toothbrush is aiming to achieve the following objectives: To retain the 20% existing loyal customers To increase current awareness level of brand from 5% to 15% in 12 months through integrated campaign of advertising, sales promotion and sponsorship. To acquire more customers by targeting the luxury segments of the market. Strategy Development The objective of Colgate Palmolive is to position the company as continuous market leader of oral care and Heath care, serving domestic market as well as international market. To enter the market Colgate will start with our dental professionals whom we have collaboration with in our existing various areas of operation. The company will make small adaptation upon on this product which will permit to save cost. There will be a launch of the product in selected dental professionals where the demonstration of how the product is use will be provided. Tactics As part of tactic the large advertising campaigns will be launched through promotion activities which will include advertising by posters, television, outdoor adverts, online adverts. B: CRITICALLY EVALUATE THE MARKETING PLAN, IN PARTICULAR ITS MARKETING ENVIRONMENT. Marketing environment entails the players and powers outside the marketing that affect management steadiness to build and sustain effective relationships with target customers. Prosperous Companies mostly identify the vital importance of continuously observing and adapting to the ever-changing environment. The marketing environment of a company can be categorised into two areas this include macro environment and micro environment, the macro environment include the main actors of the company involved in producing, distributing, and promoting the offering. The marketing environment analysis can provide insight of online and offline competitive environments. Using Five Forces Model, Porter explains that in any industry there are five forces that influence what happens within the industry, these five forces combine to make up the business environment. By studying the structure of dynamics between these forces, the company can discover opportunities for improving the marketing strategies. When developing company marketing plan which include the marketing, this can not be completely ruled out strategies Colgate Palmolive can identify the forces as: The Threat of new entrants There has been n o major new entrant to the industry for the past few years leaving the giant manufacturer of consumer goods like Colgate and others to dominate the scene. However, no major changes appear to be existing barriers to entry such as the economies of scale already benefited by the Colgate as a player who have been in the business for long. As a result of difficult trading conditions in the environment, capital requirements and integration costs are high due to the downturn in the increase interest rates rises. Power of supplier Tighter credit markets may lead to business disruptions for certain of the suppliers, contract manufacturers or trade customers which could, in turn, adversely impact colgate business. Even though this look impossible as the supplier will not want to loss Colgate business because of its global presence worldwide, this cannot be totally ruled out. Competitive Rivalry Significant competition in the industry could affect the business, for instance the rigorous competition around the world, especially from the bigger multinational consumer product companies who have greater resources than Colgate. The ability to compete depends on the strength of our brands and on our ability to protect our patent, trademark and trade dress rights and to defend against related challenges brought by competitors. A failure to compete effectively could adversely affect the growth and profitability of the company Limitation of the Five forces method The limitation of this method is the interaction of the forces with each other especially in relation to the cosmetic life cycle which affect the competitive forces significantly. Media/IT Interference at Colgate Palm Olive In 200 years of existence, information technology and media has assisted to develop a vibrant organization at Colgate. As the trend in IT changes over the years, it has also been reflected in every operations the organization. Starting with the ancient IBM system units of early 1950s to the first consumer computers of 1977 (Apple I, II III), Colgate has grown alongside to compliment the transition with time. In the modern era, parts of the positive impacts can be seen in bulletin board services, computer conferencing, computer output on microfilm, communication network, decision support systems, Electronic mail (e-mail), Electronic spread sheet, Telex, video chat, teleconferencing and the likes has been a major contribution of IT and media to the industry. Not to mention the advanced electronic instrumentation for packaging and transportation that we could notice at every stage of production. Marketing to a layman can now be found on twitter, facebook, youtube, text messaging, google, eposters and the likes. On a negative side, insecurity of data and records has been a major setback of modern IT and media in marketing. The major fear here lies in the future of the system whether or not it still pose similar advantages or negative significance. All the aforementioned means of marketing in the modern era has lots of impacts both positively and negatively: Positive Impacts of Media/IT Marketing speed and Convenience Modern IT/media has posed lots of advantages on how fast information can be disseminated. A SATA system has a data transfer rate of 300Mb/s (Wisegeek 2011). It indicates that a huge data of the size of a disc could be communicated to a large set of audience in less than one second. Marketing is now possible 24 hours a day, 365 days a year. Customers can now bridge the queuing problems at stores (Hollensen 2007). Cheaper dissemination routes Because of easy means of data flow, the cost of distribution has fallen considerably over time. It cots as low as 50 pence to fax a page whereas millions of pages will sent across the internet/media at almost a free distribution cost. Apart from the fact that the whole world receives the information at once and at the same quality, the impacts are the same with the impression expressed. Moreover, it is possible to search for the lowest price available for brands (Hollensen 2007). Online financial services are also possible through these means. Effective communication Media and IT has made it possible to decide on target audience, detailed information like product, availability, and pricing are provided for the people. Through the use of multimedia, the customers could have a better understanding of their needs by for example depicting the interior decorations or by showing the beauty of housing estates. Means of reducing workloads Marketing requires lots of workloads which have been reduced by adopting information technology. IT supports the full range of organizational functions and processes that delivers products and services to customers and other key stakeholders (Chaffey et al. 2000). Different functional parts of the organization can be corporately glued with the aid of this tool. Negative Impacts of Media/IT Despite the raging benefits enjoyed by the organizations using IT/Media, the following fears are yet to be addressed: Authentication Identity problem is still yet a major problem of modern marketing. The use of passwords as a safe key has been compounded by the hacking programmers. Loop holes have been created in lots of marketing links. This has posed a challenge to the future of e-marketing/media communication systems. Spam Disease Spam diseases in marketing involves filtering off of positive information from the direct sender by hackers or computer viruses Data Saturation Crashing marketing There is tendency for marketing crash of similar products from different manufacturers. Sometimes we found it difficult identifying the differences in the advertisement from similar companies. Insecurity channels Crackers often capture details rendering the marketing system unsafe for customers. Many fears have been expressed by customers on crackers capturing the credit/debit card details. This is still a threat to be investigated in future of information technology and media. References S Hollensen (2007) Global Marketing A decision-oriented approach, fourth edition pp. 499-506. Wisegeek (2011) A website on data transfer rate, [online], Available at www.wisegeek.com/what-is-data-transfer-rate.htm Assessed: 29 Mar. 11

Essay on Jealousy in Much Ado About Nothing, Othello, and Winters Tale

Jealousy in Much Ado About Nothing, Othello, and Winter's Tale    The common thread of jealousy ties together the main plots in Much Ado About Nothing, Othello, and The Winter's Tale. In each of these plays, the main conflict is centered around some form of jealousy. While jealousy is the mutual, most prominent cause for turmoil in these plays, its effects on the characters, and ultimately the plots, is different in each case. This difference has much to do with the way in which the concept of jealousy is woven into each play, and what it is intended to accomplish. In Othello, the jealousy factor is deliberately introduced by Iago, with the precise intention of destroying those whom he feels have wronged him. Since it is intentionally used with malicious intent, it has catastrophic results. Iago himself is jealous of Cassio; he feels that he should have been appointed to Cassio's position by Othello, and since he wasn't he hates both Othello and Cassio. Iago channels the jealousy that Othello and Cassio have made him feel, and uses it against them in a hateful plan. Iago starts the process by planting the seeds of jealousy in Othello's mind, telling him Desdemona has been unfaithful. He then proceeds to cultivate the growing jealousy by feeding it with more lies, and twisting innocent events into situations which would serve his needs (his telling Othello that Cassio and Desdemona met in secret, and convincing him that Desdemona vied for Cassio's reinstatement as lieutenant because she loved him, for example). When the seeds had flourished, and Iago had succeeded in driving Othello mad with jealousy, Iago harvested his crop and convinced Othello to kill Desdemona. Othello's killing Desdemona would both rid Iago of Desde... ...l effects, and when the mistake of jealousy was revealed the problem was solved and every one could be happy. In each of these plays, jealousy is used as a means of producing a conflict and creating trouble in the lives of the characters. The jealousy in each play, although it is introduced in a different way, always involves a man being jealous of his wife (or fiancà ©e, in Hero's case) being unfaithful with another man. Whether he misinterpret something he sees, or believe slanderous lies, the man's jealousy builds until it forces him to do something to punish his unfaithful woman. At the end of each play, the man is made to realize his mistake, but sometimes the damage can not be undone. Jealousy is the main crisis in each type of play - tragedy, trage-comedy, and comedy - but its results lie strictly in the way it is introduced, and the intended severity.   

Definition Essay - Can Love be Defined? -- Expository Definition Essay

Can Love be Defined? The other day I was babysitting my three-year-old niece, a most conniving little angel. As she sat gawking at my girlfriend's brother, Matthew, who was eating potato chips, she told me that she loved me "so much." She had already devoured her potato chips, but she obviously wanted more. Many more expressions of love proceeded to drip from her lips. Finally, the question came; "Reg, can I have some more chips?" At first, I thought this little show of bribery was cute and funny, but then I started to think about the true meaning of love. What is true love? Poets, philosophers, religious leaders, and the American media all have different definitions for this word. Too often, love is conceived as doing whatever it takes to get your potato chips. However, true love involves much more than personal satisfaction. Our society uses the term "love" far too lightly. Having been guilty of this offense myself, I admit my guilt. I often say, "I love Mexican food," or, "I love my truck." Love, however, is not just a happy feeling we get when something nice pleases us. Love is a verb. Love is the action of giving yourself totally to one person without expecting anything in return. Love is being willing to put aside your own needs in order to meet the needs of others. Giving up your weekend trip to the beach in order to take care of your sick mother is one example of love. A destitute mother giving up her one piece of bread to feed her sick little boy is another example of love. The ultimate expression of love is being willing to lay down your own life in the place of another. Love is giving, not getting. Too often our love is conditional. We often hear, "I will love you if you meet my needs," or, "I will love you... ...n who did just that. Two thousand years ago, on a hill outside of Jerusalem, Jesus of Nazareth was nailed to a cross. Three nails brutally pierced his hands and feet. Being a Christian, I believe that Jesus died on that cross to save me from an eternity in Hell. Jesus laid down his life in my place. This is the greatest love anyone has ever shown me, and my definition of love comes straight from that historical death. Would I lay down my life for anyone? I would consider going to the gas chamber in the place of very few people; this would be the ultimate test of my love. However, there are smaller ways that love can be shown. Washing dishes for my girlfriend, babysitting my niece, or even taking my mom out to eat are three small ways of showing my love. The next time I tell someone that I love them, I will be sure to follow through with the actions to prove it.

History of Mathematics Essay

If D is between A and B, then AD + DB = AB (Segment Addition Postulate). And segment AB has exactly one midpoint which is D (Midpoint Postulate). The midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle. Midsegment Theorem states that the segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to half the length of the third side. In the figure show above (and below), DE will always be equal to half of BC. Given ? ABC with point D the midpoint of AB and point E the midpoint of AC and point F is the midpoint of BC, the following can be concluded: Since the tangent of circle is perpendicular to the radius drawn to the tangency point, both radii of the two orthogonal circles A and B drawn to the point of intersection and the line segment connecting the centres form a right triangle. If and are the equations of the two circles A and B, then by Pythagorean theorem, is the condition of the orthogonality of the circles. A Saccheri quadrilateral is a quadrilateral that has one set of opposite sides called the legs that are congruent, the other set of opposite sides called the bases that are disjointly parallel, and, at one of the bases, both angles are right angles. It is named after Giovanni Gerolamo Saccheri, an Italian Jesuit priest and mathematician, who attempted to prove Euclid’s Fifth Postulate from the other axioms by the use of a reductio ad absurdum argument by assuming the negation of the Fifth Postulate. In hyperbolic geometry, since the angle sum of a triangle is strictly less than radians, then the angle sum of a quadrilateral in hyperbolic geometry is strictly less than radians. Thus, in any Saccheri quadrilateral, the angles that are not right angles must be acute. Some examples of Saccheri quadrilaterals in various models are shown below. In each example, the Saccheri quadrilateral is labelled as ABCD, and the common perpendicular line to the bases is drawn in blue. For hundreds of years mathematicians tried without success to prove the postulate as a theorem, that is, to deduce it from Euclid’s other four postulates. It was not until the last century or two that four mathematicians, Bolyai, Gauss, Lobachevsky, and Riemann, working independently, discovered that Euclid’s parallel postulate could not be proven from his other postulates. Their discovery paved the way for the development of other kinds of geometry, called non-Euclidean geometries. Non-Euclidean geometries differ from Euclidean geometry only in their rejection of the parallel postulate but this single alteration at the axiomatic foundation of the geometry has profound effects in its logical consequences. The Lobachevsky geometry is therefore consists of these statements: ? There are lines that are parallel which are everywhere equidistant. ? In any triangle the sum of the three angles is two right angles which is 180 degrees. ? Straight lines parallel to the same line are parallel to each other. ? There exist geometric figures similar with same shape but of different size to other geometric figures. ? Given three points, there is a circle that passes through all three. ? If three angles of a quadrilateral are right angles, then the fourth angle is a right angle. ? There is no triangle in which all three angles are as small as we please. ? There exist squares or equilateral quadrilaterals with four right angles.