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Feature Article: Leadership in Requisite Organization: A Spiral Dynamics Perspective
Feature Articles / January 2007
Herb Koplowitz
This paper has two purposes. First, it is an exploration and explanation of leadership as it is understood within Requisite Organization (Jaques 1996). RO (Requisite Organization) is a total systems approach to designing, staffing and managing the workplace. Most people who work in organization design or organization development have not heard of it or of Elliott Jaques who was its major designer. Jaques is so unknown, in fact, that there is a small literature on why he is unknown (Harvey, 1999) and his theory is so hard to understand (Solaas 2003). Among those in the field who do know of RO, many dismiss the approach because they find it too hierarchical, unrealistically dry or impractical. Yet there are dozens of companies that have successfully grounded their design and management in RO principles and hundreds of practitioners around the world who find that RO helps them make sense of the workplace and its problems, gives them powerful insights into how things work and powerful solutions to workplace problems. A recent conference of academics, business people and consultants drew 140 people from 11 countries and its follow up this year is expected to draw twice as many people from more countries. It takes a 520-page annotated bibliography to summarize the research that has been done on RO (Craddock 2007). Part of my first purpose is to explain why opinions of RO are so split.
The second purpose of the paper is to use this exploration of the meaning of "leadership" as a demonstration of an approach to definition of terms. "Leadership", like much of the vocabulary of organizations, has clearer connotations than denotation. Read the management advice column in your local newspaper or comments in listserves or listen in on conversations about the workplace and you will get the impression that managers are stodgy, mechanistic, and autocratic attempting only to do things right while leaders are inspiring, inclusive and exciting, attempting to do the right things. While it is clear that our field considers it better to be a leader than a manager, none of these connotations tells us exactly what a leader or a manager is.
A shared understanding of the meaning of commonly used terms, the denotation of the words, is a prerequisite for dialogue. Ask any civil engineer from anywhere in the world what work is and they will say it is force applied over a distance and they all have the same understanding of what that means. But ask 30 management consultants or organizational psychologists what a leader is and you will get 30 answers and there will be significant differences among them. We cannot meaningfully discuss whether leaders must be inspirational unless we have a shared understanding of the meaning of "leader". In well-established disciplines like civil engineering, dialogue is possible because the technical terms all have shared univocal definitions. Authors of articles in physics or civil engineering journals do not have to define terms like "force" or "work" because anyone reading the article will understand those terms as the author intends them. In our field, however, when authors use terms like "leader" and "manager" without defining them, the reader's understanding of the article may be very different from what the author intended. (And our authors rarely do define their terms.)
Given the lack of consensus aboutleadership, there is no way in which one definition of "leadership" can be said to be "correct" or "more accurate" than another. It is, of course, meaningful to ask which of two definitions is more useful under certain circumstances. But to engage in dialogue about such a question we must first be clear about the meaning each of us gives the terms we use if those terms are in danger of being significantly ambiguous. The second purpose of this paper is to conduct an exercise in comparison of such definitions.
I shall explain the RO understanding of "leadership", compare it with two other definitions and relate them to the contexts they operate in. The contexts will be viewed through two frameworks, Spiral Dynamics (as expressed by Don Beck 1996) and Requisite Organization (Jaques 1996; Jaques & Cason 1994). The exploration will take this path:
Three definitions and their origins and purposes
Description of RO and the role of leadership in it
Description of Spiral Dynamics and its implications for RO and leadership
Conclusion and implications
1. Three Definitions of Leadership
Within RO, Jaques gives "leadership" the following definition:
Leadership is setting a vision and coordinating the work of others in achieving it.
I will compare it to two others.
Joseph Rost's
"Leadership is an influence relationship among leaders and collaborators who intend real changes that reflect their mutual purposes." (Rost 1991)
Russ Volckmann's
"A leader is an individual who draws on the skills required to be self-aware, to self-manage, to comprehend the state of attunement between her own beliefs and what she perceives in the culture, to understand the relationship between her own behavior and her perceptions of the structures, systems and technologies of her organization or institution and who commits herself to the evolution of her own worldview by taking action with others to create real change." (Volckmann, 2006).
Jaques's definition often elicits negative responses:
"Leaders inspire, and there is no inspiration in this definition."
"Leadership is warm but this is cold."
"This is management, not leadership." (This comment is often made without the offer of alternative definitions of "manager" and "leader".)
His definition appears different from the other two in several ways:
In contrast to both alternate definitions, Jaques's has no reference to "real" or "significant" change and no explicit reference to change of any kind. Also, Jaques's definition has connotations of the leader's doing something to followers (coordinating their work) while the other two definitions connote being together with others ("collaborators" for Rost, "taking action with others" for Volckmann).
In contrast to Rost's definition, Jaques's lacks any reference to influence or mutual purpose.
In contrast to Volckmann's, it lacks reference to self awareness or self management, relationship between the leader and the culture or organization they operate in, and development of one's worldview
I would like to show that these three definitions are more compatible than they might seem. The major difference between Jaques's definition and the others' is his assumption of certain requisite conditions. I will next describe those conditions and then show how they affect the need for and exercise of managerial leadership. Jaques considered that there was no "leader" role, that there is no role whose sole accountability is to exercise leadership. Leadership is always an accountability of a role that has additional accountabilities. Leadership in different contexts may require different practices. This paper focuses on managerial leadership.
2. Intro to Requisite Organization
The Three Levels of Requisite Organization
Jaques's organizational engineering describes the requisite conditions for managerial leaderships. Engineering develops templates (e.g. regarding the difference in complexity between an employee's role and their manager's role) and methods (e.g. ways of assessing capability) that are grounded in science which, in turn, is structured by a conceptual framework. These are the same three levels comprising any complete discipline such as medicine or civil engineering. (Koplowitz 2007, Solaas 2007) I will explain what those three levels are and give some examples of the content of RO at each level.
The conceptual framework is the system of concepts we use to make sense of and describe the issues addressed by the discipline, in this case, the work place. Our conceptual framework influences what we pay attention to and how we diagnose organizational problems. It is the conceptual aspect of what Kuhn (1962) called a "paradigm". Following are some elements of RO's conceptual framework most relevant to the discussion of leadership:
Complexity of work in a role. One role can be said to require more complex work than another and the complexity of the work in a role can be measured.
Cognitive capacity of a person. We can say of two people that one of them can do more complex work than the other and we can assess the level of complexity an individual can handle. We call this property of a person their "cognitive capacity," also referred to as capacity for information processing (CIP) and mental complexity.
Responsibility vs. accountability. Your holding me accountable is different from my feeling responsible.
Capability. An individual's capability to work is determined by their cognitive capacity, skills and knowledge and their values (motivation—this assumes that the individual is not hindered by alcoholism, uncontrollable rage, crippling anxiety, etc.).
The conceptual framework determines what data we look for and how we make sense of the data we observe. It is the lens through which we view organizations and people at work in them. Because of this, we tend not to change our conceptual frameworks easily. If, for example, it does not make sense to you that one person has more potential than another, if you understand performance to result solely from skills and determination, no data will convince you to accept the concept of cognitive capacity. Those data will be assimilated to and interpreted by your own concept of capability. In Kuhn's term, RO requires some people to operate from a new paradigm. Likewise, at this point it would take a paradigm shift for me to not see cognitive capacity as a dimension of a person.
Science is the system of facts and causal relationships that describes objective reality, the aspects of reality we all see in the same way. You and I may disagree about how beautiful a painting is, but we should agree on how long and wide it is. And we should equally observe that that a metal rod gets longer as it gets hotter. These objective aspects of reality are what science relates to and the conceptual framework gives us the terms that the science is expressed in. Here are some of the facts and causal relationships in RO's science that will be useful in the discussion of leadership:
Complexity is discontinuous. That is, one role may require work qualitatively more complex than another. This allows us to stratify roles from Stratum I at which adult-level work begins, to Stratum VIII, the work of the CEO of a super-corporation (IBM, Exxon-Mobil, etc.).
Capability is discontinuous. People generally accept, value and benefit from guidance and ongoing accountability more from someone a stratum more capable than themselves than from someone at their own level.
People will fail at work beyond their own level of capability and will generally be bored in work at below their own level of capability.
Cognitive capacity matures at predictable rates. No means has been found of accelerating its development.
Treating people with disrespect lowers trust.
These are all testable propositions and have significant implications for leadership.
Engineering constructs a system of methods and templates for solving real-life problems, in this case, the problems of effectively pursuing strategic targets in an efficient and trust-building manner. In contrast to craft, which is grounded in precedent, engineering is grounded in science (Koplowitz 20076, Solaas 2007). Each RO template is justified by scientific facts or causal relationships.
Define "employee" as someone who, in exchange for a salary, is accountable for a) working with full commitment on tasks assigned by their manager b) giving their manager their best advice and c) staying within policy: If you make the employee accountable for their own output (e.g. by giving them bonuses for their output) you make it difficult for their manager to get them to do work that is not bonused. This definition makes the employee optimally useful as a resource for their manager.
Define "manager" as an employee accountable for a) their subordinates' outputs and working behaviour, b) developing a team of capable subordinates, c) exercising leadership (as defined above) and d) continuous improvement. Subordinate roles are a resource given to a manager, so the manager is accountable for the value the organization gains from that resource. (These four accountabilities define the term "manager" in RO.)
Give managers the authority to a) hold subordinates accountable for working on assigned tasks, b) rate the subordinate's effectiveness and assign them a merit bonus within policy, c) veto the appointment of an employee as a subordinate of theirs and d) initiate removal of a subordinate from their team. Without giving managers these authorities it is unreasonable to give them the accountabilities described above.
Give each employee a manager one stratum above them: A manager at the same stratum as the subordinate will not be able to add managerial value to the subordinate and the subordinate may resent taking direction from someone who feels more like a peer. A manager two or more strata above may become impatient with the subordinate and the subordinate may have difficulty understanding the manager.
Give each employee one manager: An employee with more than one manager will be pulled in two directions and will lose trust in the organization.
Establish and enforce a policy requiring all employees to treat each other with respect.
Leadership is exercised through nine managerial leadership practices. Five of them are in engagement with individual subordinates: 1) selection and integration of new employees, 2) assigning tasks and adjusting them as circumstances change, 3) monitoring subordinates' work and coaching them on it to become better in their current role (mentoring, understood as working with an employee regarding their future in the organization, is conducted by the MoR—manager once removed, the manager's manager—who also ensures that the manager practices good management). 4) assessing personal effectiveness and assigning merit bonuses within policy, and 5) deselection and dismissal. The remaining four practices are in engagement with the team: 1) context setting, 2) team building, 3) team planning, and 4) continuous improvement (Jaques 1996, pp. 99 – 114).
All of these practices are conducted in dialogue. That is, the employee is accountable to give their manager their best advice and the manager is accountable to consider the advice; to do otherwise would be to assume that the employee has nothing of value to add to the decision, and that would be disrespectful. But the decision is then made by the manager; if the manager is required to achieve any type of consensus before making a decision, it would be unfair to hold the manager to account for their subordinates' output.
Requisite Organization offers templates and methods for design of structure, staffing, talent pool development, team work, compensation, etc. and all of the templates and methods are compatible with each other. The system is clearly hierarchical and designed to maximize organizational effectiveness. At the same time, it maximizes opportunity for employees and ensures that they are treated with respect. I next wish to show how such a system affects the need for and exercise of leadership. It should be apparent that the various pieces of the RO model are interdependent with each other. As Harald Solaas (2003) said, "Good managerial leadership is not the effect of something individuals have, either innate or acquired, but of a managerial system." That entire system is rarely fully in place.
Leadership in the Context of Requisite Organization
In Section 1 I introduced the following three definitions:
Jaques: leadership is setting a vision and coordinating the work of others in achieving it.
Rost: leadership is an influence relationship among leaders and collaborators who intend real changes that reflect their mutual purposes.
Volckmann: A leader is an individual who draws on the skills required to be self-aware, to self-manage, to comprehend the state of attunement between her own beliefs and what she perceives in the culture, to understand the relationship between her own behavior and her perceptions of the structures, systems and technologies of her organization or institution and who commits herself to the evolution of her own worldview by taking action with others to create real change.
I will now review the differences observed between Jaques's definition and the other two but interpret them in light of his assumption of requisite conditions. It will be clear that the requisite conditions are not always in place, even in organizations that attempt to use RO templates and methods. But it is the assumption of these conditions that makes the three definitions much more compatible than they at first appear to be.
Jaques's definition has no reference to "real" or "significant" change and no explicit reference to change of any kind.
In the requisite context, a manager is accountable for continuous improvement, so change is an implicit part of managerial leadership. And in that context, it is expected that employees will be given tasks that make use of the full extent of their capabilities, so managerial leaders will be asked to make changes as significant and as real as they are capable of making.
Jaques's definition has connotations of the leader's doing something to followers (coordinating their work) while the other two definitions connote being together with others ("collaborators" for Rost, "taking action with others" for Volckmann).
RO is clearly hierarchical, and employees lack the accountabilities and authorities their managers have, but this in no way prohibits collaboration. To begin with, employees are in dialogue with each other and with their manager. Also, in building teamwork, the manager – after dialogue with subordinates – determines how the work of each subordinate interacts with the work of the others. This clarity facilitates better collaboration.
Jaques's definition lacks any reference to influence or mutual purpose.
Jaques had earlier defined leadership as the ability (or accountability) "to influence one or more others – followers – willingly to accept the leader's purpose and goals and all to move in the direction set by the leader by suffusing authority with leadership practices appropriate for that role" (Jaques 1996, p. 95). He changed his definition on realizing that the managerial leader has no need to influence subordinates as they already agreed, on becoming employees, to work on any task assigned them by their manager. Employees must value the work they do or they will not do it as well as they can, so a mutual purpose is also an assumed condition; the manager is engaging subordinates in work they want to do.
Jaques's definition lacks reference to self awareness or self management
RO does not guarantee managerial leaders with advanced social skills nor with exceptional insight into themselves. But it does aim for managers who can add value to their subordinates' work, who held accountable for developing their subordinates and for treating them with respect. The manager-subordinate relationship is intended to be supportive without being intrusive. I believe the requisite conditions bring about the support that self awareness and self management are expected to result in.
Jaques's definition lacks reference to development of one's worldview
Every manager has a manager who coaches them to be better in their current role. Every manager has an MoR (manager once removed, manager's manager) who mentors them to be better prepared for future roles. The expectation is that managers' world view will be expanding
The intentions of the various definitions are not as far apart as they might appear. The difference lies more in the assumptions made about the context in which leadership is exercised.
3. Spiral Dynamics and Its Implications For RO and Leadership
Why would Jaques assume conditions that others do not assume? Why do some practitioners find the RO model so appealing while others find it disturbing? I got my first clue to this puzzle some time in the early 1990's when Brian Brittain, a colleague of mine, attended a half-day introduction to Requisite Organization given by Jaques. As people filed out of the session at the end, Brian observed "older men in dark suits who loved everything they had heard, for all the wrong reasons, and younger women who hated everything they heard, for all the wrong reasons." What he meant was that the men mistakenly heard in Jaques's defence of hierarchy support for their own rather brutal style of management. The women, whom he believed valued collegiality and respect, made the same mistake. They assumed that Jaques approved the type of hierarchy they had experienced and rejected.
Don Beck (Beck and Cowan 2006) has made a study of the kinds of values, norms and beliefs that separated those two groups in Jaques's lecture. His Spiral Dynamics identifies distinct stages of development that cultures go through whether they be national cultures or cultures of work organizations or of families. As a group meets more complex challenges, it must attain better coordination among its members. This requires changes of beliefs, values and behavioural norms. The development of a culture is not continuous. Certain constellations of beliefs, values and norms coalesce and work harmoniously with each other and define a stage of development. A group will not move out of its current stage unless challenged to by its environment. Authors typically identify eight or nine stages of cultural development, but I would like to focus on three of them, colour coded as orange, green and yellow (the colours have no special significance other than that warm colours are used to refer to cultures focused on the individual while cool colours, alternating with the warm, refer to cultures focused on the collective) the fifth, sixth and seventh stages. I believe that the assumptions made about leadership are, in effect, assumptions made about stage of cultural development.
I shall draw largely on Mehltretter and Carter (2007) in describing these three stages and their relationship to leadership.
Orange, Green and Yellow
The three stages are illustrated briefly in Table 1 below.
Orange, the fifth stage, is "interested in better, faster, cheaper" as manifested by "the total quality movement of the 80's and 90's. The tools and templates found in this model embrace bottom line results, continuous improvement, and increasing quality. Management by Objectives, In Search of Excellence, and the One Minute Manager are consistent with …orange."
These orange values have a healthy place within organizations, but can become unhealthy when overdone. When the "crunching numbers" becomes the sole focus of an organization, employees become just another resource to be used and discarded in an effort to reach "the goal". When bonus programs and commission sales systems begin to pit one employee or department against another, the system can become toxic."
The focus in orange is on the individual, e.g. the objectives an individual is to work towards. There is little in such a system to foster collaboration, dialogue or respect. It would take a leader of exceptional self awareness and self management to resist the pull of the culture and actually bring about collaboration and respect for employees.
Eventually, orange leads to burnout. In reaction to this, executives work "with coaches to learn how to process their emotions, to negotiate win-win solutions, to become sensitive to the needs and desires of others…[They are] called upon to be stewards and servants, not movers and shakers. Organizational development consultants …call for dialogue, conflict resolution training, and appreciative inquiry as a means for enhancing communication, facilitating sharing, and creating learning organizations. Employees …[are] called upon to be open, cooperate, and collaborate….However, a parallel reality is in play during this harmonious time. …[In green],…systems and structure are generally either downplayed or completely dismantled. This hard-won freedom eventually becomes a huge liability in the green organization as it struggles to balance intense competition, caring for its people, managing its business, interacting with partners, and vendors all at the hands of long-atrophied systems. Eventually green organizations find that the cost of caring for people becomes too great."
The green values of collaboration and respect are advances over the orange competitiveness. But a green culture may turn respect into political correctness and denial of real and significant differences. The focus on the collective may prevent individual accountability which is required for the next stage of coordination. But it is not at all surprising that individual accountability, which implies hierarchy, would be resisted by people whose only understanding of hierarchy is as it was practiced in orange.
The yellow stage accepts the inevitability of change. It requires big picture views and integrative structures; teamwork cannot be expected to happen freely of its own accord at this level of complexity. While there is respect for the individual, individual differences are recognized. And while collaboration is valued, it is achieved through individual accountability.
This, I believe, is the stage that Jaques had in mind when he described RO. Those who expect orange or green cultures will be suspicious of it. After all, orange will reject it when it recognizes it because it requires too much respect for the individual and too much planning of integrative systems. And green will reject it because it looks like orange to them.
Leadership In a Spiral Dynamics Context
The developmental stage of a culture determines not only how leadership is practiced but also how it is described and defined. When comparing definitions of "leadership" it becomes useful to examine the cultural and systems assumptions behind the definition.
4. Conclusion and Implications
If we were studying leadership within a wolf pack or a herd of elephants to determine significant characteristics of the leader, we would be engaged in science. We would be describing objective reality, what anyone would observe when watching the lead wolf or elephant.
But we are concerned about leadership in organizations and other social groups. We are concerned with design, and that is an engineering activity. We are using psychology, what we understand of human nature, as a basis for designing templates and methods to solve real-life problems. Defining "leadership" may seem an academic exercise. But when the CEO asks how the organization is to attract and develop leaders, it is useful to know what we are attracting and developing. In defining "leadership", we are seeking not truth but usefulness.
Beck, D., and Cowen, C. (1996, 2006). Spiral dynamics: Mastering values, leadership, and change. Malden, M A: Blackwell.
Craddock, K. (2007). RO Bibliography. http:// www.globalro.org. Accessed January 9, 2007.
Harvey, J. (1999)."Musing About the Elephant in the Parlour"Who the Hell is Elliott Jaques?", pp. 173 – 202 in Harvey, Jerry How Come Every Time I Get Stabbed In the Back My Fingerprints Are On the Knife? Jossey-Bass.
Jaques, E. (1996). Requisite Organization: A Total System for Effective Managerial Organization and Managerial Leadership for the 21st Century. Arlington, V A: Cason Hall and Co., Publishers.
Jaques, E. and Cason, K. (1994). Human Capability: A Study of Individual Potential and its Application. Falls Church, V A: Cason Hall and Co., Publishers.
Koplowitz, H. (2007). "How Solid Is Requisite Organization?" in Shepard, Ken, Jerry Gray and Jerry Hunt (Eds.) The Executive's Guide to Organization Design, Levels of Work and Human Capability (working title).
Kuhn, Thomas (1962). The Structure of Scientific Revolutions. Chicago: The University of Chicago Press.
Mehltretter, Glenn W., Jr. and Michelle Malay Carter (2007) "Assessing Requisite Readiness" in Shepard, Ken, Jerry Gray and Jerry Hunt (Eds.) The Executive's Guide to Organization Design, Levels of Work and Human Capability (working title).
Rost, Joseph C. (1991). Leadership for the Twenty-First Century. New York: Praeger.
Solaas, H. (2003). "Why RO theory is so difficult to understand" (unpublished paper).
Solaas, H. (2007). "Organizational Engineering and the CEO" in Shepard, Ken, Jerry Gray and Jerry Hunt (Eds.) The Executive's Guide to Organization Design, Levels of Work and Human Capability (working title).
Volckmann, R. (2006). "Definition of Leadership: An Integral Agenda" Submitted for publication.
Herb Koplowitz, PhD, is President of, Applied Organizational Science. He has worked with clients in the utility, pharmaceutical, financial, high tech, education, food, and automotive retail sectors as well as the Canadian Armed Forces, helping them with their structure, performance management and talent pool development. Herb bases his work in Requisite Organization. He has developed training materials and consulting approaches that translate Requisite theory into step-by-step practices. Herb makes Requisite theory available to those whose major strategic focus is not on management by applying Requisite principles to the issues of a particular client or group of clients. Herb works with consultants and executives for whom good structure and good management are themselves part of strategy. He has taught Requisite Organization to business leaders through York University and to consultants through ACCORD. Herb is trained in Requisite Organization and marketing research. He holds a B. A. in mathematics and philosophy from Cornell University, a Ph.D. in psychology from the University of Massachusetts and registration as an organizational psychologist in Ontario.
E-mail herb@ca.inter.net
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You are at:Home»News»Asia News»Hong Kong protesters hold rally for the Uighur minority in China
Hong Kong protest rally for the Uighurs (Photo: YouTube)
Hong Kong protesters hold rally for the Uighur minority in China
By Froilan Malit on December 23, 2019 Asia News, News
Waving Uighur flags and posters in the city's financial district, more than 1,000 people in Hong Kong rallied in support of China's ethnic Uighurs on Sunday, December 22, marking the latest demonstration in over six months of anti-government unrest.
A mixed crowd of young and elderly people, dressed in black and wearing masks to shield their identities, held up signs reading "Free Uyghur, Free Hong Kong" and "Fake 'autonomy' in China results in genocide".
"I think basic freedom and independence should exist for all people, not just for Hong Kong," said a 41-year-old woman surnamed Wong who attended the protest with her husband.
Clashes later broke out between Hong Kong police and the protesters, with police arresting two protesters who were attempting to burn a Chinese flag at the rally.
United Nations experts and activists say at least 1 million Uighurs and members of other largely Muslim minority groups have been detained in camps in Xinjiang since 2017 under a campaign that has been condemned by the United States and other countries.
The protest comes after midfielder Mesut Ozil of English football club Arsenal caused a furore in China after he criticised the country's policies toward the Muslim ethnic minority in the restive northwestern region of Xinjiang.
Ozil, a German Muslim of Turkish origin, tweeted that Uighurs were "warriors who resist persecution" and criticised both China's strong hand in Xinjiang and the relative silence of Muslims in response.
Beijing says it is providing vocational training to help stamp out separatism and to teach new skills. It denies any mistreatment of Uighurs.
Protests in Hong Kong are now in their seventh month, albeit in a relative year-end lull. Many residents are angry at what they see as Chinese meddling in the freedoms promised to the former British colony when it returned to Chinese rule in 1997.
China denies interfering and says it is committed to the "one country, two systems" formula put in place at that time and has blamed foreign forces for fomenting unrest.
On Saturday, Hong Kong riot police swept into several shopping malls, chasing off and arresting demonstrators pressing their demands in the peak shopping weekend before Christmas.
The police's heavy-handed clamp-down on demonstrations and frequent use of tear gas have incensed many protesters, whose demands include full democracy and an independent investigation into perceived police brutality.
The protests, which started in June, have pushed Hong Kong's economy into recession. Retailers and businesses have been hit hard as tourists stay away amid transport disruptions.
The city's Financial Secretary Paul Chan, in his blog on Sunday, said there could be a wave of business closures in the new year if market conditions do not improve.
"Many companies are now clenching their teeth and struggling to maintain their businesses," he said, adding that he hoped the violence would stop. (Source: CNA/AP)


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they form part: but these circumferences are to each other as their radii AC, DO ; hence
arc AB
arc DE :: AC : DO.
For a like reason, the sectors ACB, DOE are to each other as the whole circles; which again are as the squares of their radii; therefore
sect. ACB sect. DOE :: AC2: DO2.
PROPOSITION XII. THEOREM.
The area of a circle is equal to the product of its circumference by half the radius.
Let ACDE be a circle whose centre is O and radius OA: then will
area OA=10A × circ. OA.
For, inscribe in the circle any E regular polygon, and draw OF perpendicular to one of its sides. Then the area of the polygon will be equal to OF, multiplied by the perimeter (Prop. IX.). Now, let the number of sides of the polygon be indefinitely increased by continually bisecting the arcs which subtend the sides: the perimeter will then become equal to the circumference of the circle, the perpendicular OF will become equal to OA, and the area of the polygon to the area of the circle (Prop. VIII. Cor. 1. & 3.). But the expression for the area will then become
Cor. 1. The area of a sector is equal to the arc of that sector multiplied by half its radius.
For, the sector ACE is to the whole circle as the arc AMB is to the whole circumference ABD (Book III. Prop. XVII. Sch. 2.), or as AMB× AC is to ABD × AC. But the whole circle is equal to ABD x AC; hence the sector ACB is measured by AMB × AC.
area OA=10A × circ. OA : consequently, the area of a circle is equal to the product of half the radius into the circumference.
Cor. 2. Let the circumference of the circle whose diameter is unity, be denoted by : then, because circumferences are to each other as their radii or diameters, we shall have the diameter 1 to its circumference, as the diameter 2CA is to the circumference whose radius is CA, that is, 1 : 2CA: circ. CA, therefore circ. СA=л× 2CA. Multiply both terms by CA; we have CA x circ. CA =л× СА2, or area CA: =лX CA2: hence the area of a circle is equal to the product of the square of its radius by the constant number, which represents the circumference whose diameter is 1, or the ratio of the circumference to the diameter.
In like manner, the area of the circle, whose radius is OB, will be equal to π × OВ2; but л× СА2 : π × OВ2 : : CA2 : OB2 hence the areas of circles are to each other as the squares of their radii, which agrees with the preceding theorem.
Scholium. We have already observed, that the problem of the quadrature of the circle consists in finding a square equal in surface to a circle, the radius of which is known. Now it has just been proved, that a circle is equivalent to the rectangle contained by its circumference and half its radius; and this rectangle may be changed into a square, by finding a mean proportional between its length and its breadth (Book IV. Prob. III.). To square the circle, therefore, is to find the circumference when the radius is given; and for effecting this, it is enough to know the ratio of the circumference to its radius, or its diameter.
Hitherto the ratio in question has never been determined except approximatively; but the approximation has been carried so far, that a knowledge of the exact ratio would afford no real advantage whatever beyond that of the approximate ratio. Accordingly, this problem, which engaged geometers so deeply, when their methods of approximation were less perfect, is now degraded to the rank of those idle questions, with which no one possessing the slightest tincture of geometrical science will occupy any portion of his time.
Archimedes showed that the ratio of the circumference to the diameter is included between 34% and 341; hence 34 or 27 affords at once a pretty accurate approximation to the number above designated by ; and the simplicity of this first approximation has brought it into very general use. Metius, for the same number, found the much more accurate value 355. At last the value of л, developed to a certain order of decimals, was found by other calculators to be 3.1415926535897932, &c.;
and some have had patience enough to continue these decimals to the hundred and twenty-seventh, or even to the hundred and fortieth place. Such an approximation is evidently equivalent to perfect correctness: the root of an imperfect power is in no case more accurately known.
The following problem will exhibit one of the simplest elementary methods of obtaining those approximations.
PROPOSITION XIII. PROBLEM.
The surface of a regular inscribed polygon, and that of a similar polygon circumscribed, being given; to find the surfaces of the regular inscribed and circumscribed polygons having double the number of sides.
E PM Q
Let AB be a side of the given inscribed polygon; EF, parallel to AB, a side of the circumscribed polygon; C the centre of the circle. If the chord AM and the tangents AP, BQ, be drawn, AM will be a side of the inscribed polygon, having twice the number of sides; and AP+PM 2PM or PQ, will be a side of the similar circumscribed polygon (Prop. VI. Cor. 3.). Now, as the same construction will take place at each of the angles equal to ACM, it will be sufficient to consider ACM by itself, the triangles connected with it being evidently to each other as the whole polygons of which they form part. Let A, then, be the surface of the inscribed polygon whose side is AB, B that of the similar circumscribed polygon; A' the surface of the polygon whose side is AM, B' that of the similar circumscribed polygon: A and B are given; we have to find A' and B'.
First. The triangles ACD, ACM, having the common vertex A, are to each other as their bases CD, CM; they are likewise to each other as the polygons A and A', of which they form part hence A: A':: CD: CM. Again, the triangles CAM, CME, having the common vertex M, are to each other as their bases CA, CE; they are likewise to each other as the polygons A and B of which they form part; hence A': B:: CA CE. But since AD and ME are parallel, we have CD: CM:: CA: CE; hence A : A':: A': B; hence the polygon A', one of those required, is a mean proportional between the two given polygons A and B and consequently A'= √ A× B.
E P M Q
Secondly. The altitude CM being common, the triangle CPM is to the triangle CPE as PM is to PE; but since CP bisects the angle MCE, we have PM: PE : : CM : CE (Book IV. Prop. XVII.):: CD CA :: A : A': hence CPM: CPE :: A: A'; and consequently CPM: CPM+ CPE or CME : : A : A+ A'. But CMPA, or 2CMP, and CME are to each other as the polygons B' and B, of which they form part: hence B': B : : 2A : A+A'. Now A' has been already determined; this new proportion will
2A.B
A+A, and thus by means of the polygons A and B it is easy to find the polygons A' and B', which shall have double the number of sides.
serve for determining B', and give us B'
PROPOSITION XIV. PROBLEM.
To find the approximate ratio of the circumference to the diameter.
Let the radius of the circle be 1; the side of the inscribed square will be √2 (Prop. III. Sch.), that of the circumscribed square will be equal to the diameter 2; hence the surface of the inscribed square is 2, and that of the circumscribed square is 4. Let us therefore put A=2, and B=4; by the last proposition we shall find the inscribed octagon A'=√8—2.8284271, and the circumscribed octagon B'=; inscribed and the circumscribed octagons being thus determined, we shall easily, by means of them, determine the polygons having twice the number of sides. We have only in this case to put A=2.8284271, B=3.3137085; we shall find A'= 2 A.B A+A
2+8=3.3137085. The
✓A.B=3.0614674, and B'= =3.1825979. These polygons of 16 sides will in their turn enable us to find the polygons of 32; and the process may be continued, till there remains no longer any difference between the inscribed and the circumscribed polygon, at least so far as that place of decimals where the computation stops, and so far as the seventh place, in this example. Being arrived at this point, we shall infer
that the last result expresses the area of the circle, which, since it must always lie between the inscribed and the circumscribed polygon, and since those polygons agree as far as a certain place of decimals, must also agree with both as far as the same place.
Number of sides
We have subjoined the computation of those polygons, carried on till they agree as far as the seventh place of decimals.
The area of the circle, we infer therefore, is equal to 3.1415926. Some doubt may exist perhaps about the last decimal figure, owing to errors proceeding from the parts omitted; but the calculation has been carried on with an additional figure, that the final result here given might be absolutely correct even to the last decimal place.
Since the area of the circle is equal to half the circumference multiplied by the radius, the half circumference must be 3.1415926, when the radius is 1; or the whole circumference must be 3.1415926, when the diameter is 1: hence the ratio of the circumference to the diameter, formerly expressed by π, is equal to 3.1415926. The number 3.1416 is the one generally used. L*


text:
USAF test launches unarmed Minuteman III intercontinental ballistic missile
The US Air Force (USAF) Global Strike Command Airmen has tested an unarmed Minuteman III intercontinental ballistic missile installed with a test re-entry vehicle from Vandenberg Air Force Base, California.
By Samseer M
The ICBM's re-entry vehicle consisted of a telemetry package used for operational testing. The vehicle travelled around 4,200 miles to the Kwajalein Atoll in the Marshall Islands.
This test launch was aimed to verify accuracy and reliability of the ICBM weapon system, providing valuable data to ensure a safe, secure and effective nuclear deterrent.
Task Force launch commander major Jacqueline Sartori said: "Operational test launches are an amazing opportunity for our airmen to see the results of their hard work first-hand.
"This launch will help us verify the reliability of our ICBM weapon system and processes."
"This test launch was aimed to verify accuracy and reliability of the ICBM weapon system."
The launch was initiated by crew members from the 625th STOS. It took place aboard the US Strategic Command Airborne Command Post E-6B aircraft.
The data collected from this mission will be used by the US Department of Defense, the Department of Energy, and US Strategic Command for continuing force development evaluation.
Under the direction of the 576th Flight Test Squadron, Vandenberg AFB, the launch team included airmen from the 90th Missile Wing at FE Warren AFB, Wyoming, and the 625th Strategic Operations Squadron at Offutt Air Force Base, Nebraska.
625th STOS commander lieutenant colonel Grayson Higby said: "The test marks the culmination of dedicated individual effort, as well as effective coordination between multiple DoD organisations."
In March this year, the USAF Global Strike Command launched an unarmed LGM-30G Minuteman III intercontinental ballistic missile (ICBM) equipped with a test re-entry vehicle.
Image: An unarmed Minuteman III intercontinental ballistic missile launches from Vandenberg Air Force Base. Photo: courtesy of US Air Force photo by Joe Davila.
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The Hill Country Care Center in Dripping Springs had some fun guests visit on Friday, Dec. 5th. AESA's Elementary classes with their teachers, Mrs. Hutton and Mrs. Patty, brought lotion and beautiful fleece blankets to the residents. The students assisted the residents with finishing the blankets as well as cheering them up. Thank you, AESA Elementary, for giving to others during the Holiday Season.


The Supreme Court on Tuesday ordered the Sandiganbayan to continue its trial of the P224-million plunder case against former Sen. Ramon "Bong" Revilla Jr. and his coaccused.
This after the Supreme Court en banc denied with finality the appeal of Revilla, his chief of staff Richard Cambe, pork barrel scam mastermind Janet Lim-Napoles, Napoles' employees Ronald John Lim and John Raymund de Asis, and Budget Undersecretary Mario Relampagos to overturn its decision two months ago denying their motion.
Revilla and his coaccused assailed the Ombudsman's resolution declaring probable cause to charge them for looting his Priority Development Assistance Fund (PDAF) by diverting the money to fake foundations and ghost projects and recipients owned by Napoles.
In its Dec. 6 decision, the Supreme Court ruled to dismiss the petition for lack of merit and affirm the Ombudsman's finding of probable cause against Revilla and his coaccused.
"(We) direct the Sandiganbayan, as the trial court, to commence or continue all necessary proceedings in these cases with deliberate dispatch," said the Supreme Court.
Napoles was charged as the "mastermind" of the PDAF scam, from the solicitation of funds to creating fake nongovernment organizations to drawing up spurious liquidation papers to giving kickbacks; her employees Lim and De Asis for acting as dummies in the bogus foundations, and Relampagos for acting as the "contact" of Napoles in the Department of Budget and Management.
Revilla has been under detention since June 2014 at the Philippine National Police Custodial Center in Camp Crame, Quezon City, along with former Sen. Jinggoy Estrada who was also charged and arrested for the pork barrel scam.
Former Sen. Juan Ponce Enrile also faced the same plunder cases and was arrested in 2015 but he was freed on bail by the Supreme Court in 2015 because of poor health and old age.


Alexey Miller told the Financial Times that the world was undergoing "a great surge in oil and gas prices . . . which will end with prices at a radically new level".
Mr Miller continued: "In the coming years Gazprom will be not just a major company in the world, but the most influential in the energy business," adding that its target was to reach a market capitalisation of $1,000bn.


I'm tearful and sad today – I have too much in my brain and too much in my heart going on and that coupled with a disrupted routine, being away from home, lack of sleep, weight gain and IBS and I am done in.
I know it will pass, I trust that now, but it doesn't stop the down days sucking and the fact that this has come while I'm on holidays and meant to be re-booting for the year ahead is not helping (although I'm aware the pressure I have on using these holidays to 're-boot' is a factor). I'm just feeling very alone in the darkness even though I have amazing support, it's too hard to explain. I'm also very guilty for feeling down even though I have a truly wonderful life. I don't want to feel like this and I'm frustrated that I can't make it go away.
None of these things cured me, but I reckon I would be feeling far worse if I haven't tried them I've also engaged in some far less helpful behaviours which made me feel better temporarily but then far worse, but let's not concentrate on them for the moment. I think I could really do with a good cry but I need time and space alone to do that and I just don't have that at the moment. I really really really miss living on my own.
I wish I could just press pause so I could roll up in the cupboard and cry, but life is too smart for that so will continue around me waiting for me to join in again. Thanks for listening.
It's holiday time! Yay! This break feels like it has been a long time coming, work and study have been hectic the last few months. I've been doing well keeping on top off things and have been careful about not working weekends etc. but that has meant long days during the week and a very full brain. The long days I can cope with mostly, especially when I'm eating enough and keeping it in, that makes a huge difference, but the 'full brain' had been starting to get to me. It means waking up in the middle of the night thinking about my to-do list, it's watching tv not paying attention to what's happening as I'm thinking about work things, it's drifting off while with friends to mentally plan where I'm at with other things. Basically, I've been the polar opposite of mindful for the last while and it had started to take its toll on my mood.
However, I'm now on holidays and trying very hard to wind down. I'm in Scotland for a week and then Ireland for a week. I've got a few days at the Edinburgh Fringe Festival which is very exciting as I love theatre, comedy, dance, anything like that and then I'm going to visit my relations further north. In Ireland then I'm going to catch up with my friends and relations but I also have a few days PhD work to do which is kind of stressing me out as I've got to do a few bit this week before I'm ready for next week. I thought it would be fine being on holidays even if I did have some work to do but it feels like it's just hanging over me now. It's great to be making progress with my PhD but it's kind of ruining my relaxation!
I'm hoping now I've put it down in words that I'm a bit stressed by that, I'll leave it here in my laptop and not carry it around in my brain all day! It is what it is, and when the work is done I will feel great about it. It's up to me whether it stresses me out all week or not. I need to draw on all my mindfulness abilities over the next few days I think, and get lost in the fun of Edinburgh. I'm lucky to be here so I need to focus on that!
PS: Thanks for the nominations and tag posts- I will get to them when I'm back home!
I had an interesting conversation with my nurse yesterday about how I seem to focus a lot on the 'micro' and can sometimes miss the 'macro' of life. We were talking about a wedding I'm going to at the weekend and she asked me what I was wearing and unbeknownst to her (but probably not a surprise to you guys!) I had a little panic about my outfit and how I was going to feel in it.
I bought a dress for this wedding (one of my best friends) a few months ago. It's elasticated so I know it will fit, but I am worried about how my stomach, arms and back will look in it. I told her that I was trying to plan ahead for what I would eat and drink, and how long I needed to run for the morning of it to feel okay in my dress (clearly I haven't quite separated the exercise/food thing: uh-oh) and what my other friends would be wearing and whether I should wear a safer dress etc. She let me ramble for a little while and then interrupted me to say that the biggest thing I had just said was that my best friend was getting married and that I was travelling to see it and celebrate with her. That's the macro, that's the important point of this weekend, that's where the living happens. Whether or not I have a bit of a stomach roll when I sit down is a micro detail in the grand scheme of things.
She understood that while I was ill and still in my ED mindset that yes, these 'micros' were a huge, extensive part of my thoughts, but she challenged me to consider moving move beyond this now. We discussed that lots of my thoughts and actions are still based on the 'micro': I don't eat sandwiches out because of the butter/mayo, I walk or cycle everywhere in case one bus trip means I put my fuel in/out sum out of balance, I choose products carefully because of what is says on the label, I obsess over if my back is too flabby in tops, I worry about how much time I spend sitting……..things like this. I have made huge strides in my eating and attitude but I am still caught up on some of these small things and they do take up a lot of my thoughts sometimes.
I think I'm afraid to let go of some of the eating things in case it's the one thing that's stopping my weight from ballooning, and some of the other ED things in case I lose all control, but she feels I'm ready to test this out now. I need to trust that if I eat three balanced meals snacks a day, exercise within reason and just wear clothes I know will fit I can free up thoughts and energy to immerse myself in the important things, like going to a wedding where I get to see my amazing friend marry the love of her life, chat with friends who I don't get to see very often, dance to cheesy pop music and enjoy myself. Doesn't that sound like far more fun than worrying what I look like in my dress?!
So this is my challenge for the next month until my next appointment. I'm going to let go a little, try not to focus on the little things and instead get caught up with living without my ED. She'll weigh me then and will hopefully show me that doing that has not impacted on my weight. I'm a bit suspicious and very nervous, but I'm committed to trying – wish me luck!
Potential trigger warning- nasty ED thoughts.
Realising my lucky dress was now tight on me was a not so nice surprise, I was flooded with ED thoughts which played over and over again throughout the day and the weekend.
Except now I've been eating a healthy amount for a while I can recognise how dangerous these thoughts can be. It doesn't matter if it's 1 kilo or 10 kilos, the moment I decide to try and lose weight is the moment I take myself off the path to recovery and head straight back to being ill. Skipping one meal here and there exposes me to starvation. Cutting back during the day leads to binges when I'm alone. Adding in an extra exercise session results in an exhaustion that carries over to daily life. I know what life with an eating disorder looks like and I don't want my future to be like that.
All of this means that I have to accept my body as it is now. I have to accept that's it rounder then I'd like. I have to be okay with the wobbles. I have to cope when clothes don't fit. I have to embrace that my body looks like a woman's. I have to learn to love the softness instead of all the sharp bony edges.
I wish with all my might that I was a naturally thin person, and that recovery for me would mean a flat stomach, skinny arms and fitting into all the clothes I want, but that is not the body I was born into and if I really want to recover I need to learn to be okay with that. It's time to make peace with my body so that I can find peace in my mind. It's been far too long, it's time to move on.
Disaster! Lucky dress no longer fits!
I have an interview tomorrow which I really should be preparing for but I just tried on my 'lucky dress' (a smart plain black dress which I have worn to pretty much all of my interviews in life!) and it's really tight. I know I shouldn't let this bother me but I am kind of panicking now-argh!
I had thought that I must try on the dress yesterday as the last time I wore it I was a fair bit smaller but then reminded myself I had worn it for an interview about 3 years before that when I was at what I thought was the same size as now. It's way tighter than it was last year, which I'm kind of okay with as I knew that would happen as I know I've put on weight and recovery is great yadda yadda yadda, but I'm horrified that it's tighter than it was three years okay. It means I'm bigger than I thought now and all the fight I've been putting into accepting my body at its 'set point' has gone flying out the window. Now instead of just worrying about my interview performance I'm stressing about looking fat and feeling horribly uncomfortable. Yuck.
I can't think of anything else to wear now and can't face trying anything on so I've given up and put on pjs and I'm hiding in bed. I know it's not the end of the world but man this recovery process is exhausting sometimes-I wish I could just step out of my body for a few days to not have to deal with it, bleurgh.
I just watched a lovely introduction video from Onepersonsinsanity who suggested that others might like to post a bit about their 'well' sides as well as their 'ill' sides. I thought this was a good idea, so although I don't like to give away too much identifying information I will give it a little go!
My name is H, I'm 33 and I currently live in the UK. I've also lived in Ireland, Scotland and Australia (and Canada, Spain and France for a few months at a time!)- I'm lucky to have family in lots of different cities and I love visiting new places. I work with children with special needs and I also teach at a University- two jobs keeps me busy but I love them both. I'm in the middle of studying for my PhD but this has been a bit on and off so it will probably be another few years before that gets done! I'm patient about it though and know it is not worth becoming sick over.
I love to run and make sure to emphasise that this is separate from my eating disorder- I do it socially with a club and a lovely group of friends and I love the fun of training and racing without being too fixed on goals and definitely not fixed on calories. I also cycle- I live near the countryside so can cycle from my house to the wilderness (well, kind of!) in only a few minutes and I know a few good routes that all pass by lovely tea shops on the way home (that's how seriously I take my cycling!). I used to sea swim a lot too when I lived by the coast- all year round despite freezing temperatures at times- I really miss that now living inland but I have to make do with a swimming pool and the occasional wild swim in a nearby river. Seeing as I swim, cycle and run it makes sense that I also do triathlons- again not in a competitive way but more that they are a great day out and a good way to see new places. Plus there's something kind of fun about struggling out of a wetsuit at speed to then jump on a bike sopping wet before trying to run with legs that feel like lead (no honestly that is fun!).
II live with my boyfriend and moving in with him was a new adventure for me this year but it's going well. I never thought I would meet someone that I would want to live with and 'settle down' with so this has been an exciting change the past couple of years. He knows about my 'issues' and while I know he would try to understand it's not something we talk about. I'm not sure whether this is the right approach or not but it's working for the moment.
I like to read although I find that when I start a book, I can't put it down so finish it really quickly, and then don't start another for a while (all or nothing ey?!). I've just read 'Still Alice' which was great and I'm about to start 'Being Mortal' as it was recommended to me by my Mum. Other than that, I spend the rest of my time hanging out with my wonderful friends and family (I am truly blessed when it comes to family and friendship), cleaning (gives me such a sense of calm) and napping (as I do love a good nap!).
Okay so it turns out writing about myself is actually quite easy- I didn't expect this to be so long! Well done if you made it to the end and I would love to read a bit more about your 'well sides' too!
A while ago I posted about clearing out my wardrobe so that when I opened it I was greeted by clothes I liked and fit me rather than being taunted by too small clothes and too big clothes and all the memories they bring. This was a great step to have made but it left me with very little to wear and a need to brave a shopping trip.
Now I'm not the best shopper anyway in that I have to be careful with money so I stress a little about that and also the getting in and out of clothes to try things on and queuing etc. makes me weary (I can run a marathon but ask me to walk around a shopping centre and I get lazy!).
I felt kind of pretty!!!!
I'm glad I persevered and I'm really reassured that I can have positive feelings about myself that are not connected to my weight/size. I felt pretty not because I looked thin or anything like that – but simply because I was wearing a nice summer dress that was comfortable and I felt okay in.
I know it may seem shallow and superficial to be pleased about liking how I look for a day, but when you spend a lifetime hating yourself in the mirror it's actually a big deal.


I work in the high tech industry, how can I participate in the new Global Talent Stream recently launched by the Canadian government?
The Global Talent Stream was launched on June 12, 2017 and is a two-year pilot under the Temporary Foreign Worker (TFW) program. The Global Talent Stream is divided into category A and B and both have a $1,000 processing fee per position, which must be paid for by the employer, and requires the employer to work with ESDC to create a Labour Market Benefits Plan.
I understand there is a new Canadian Citizenship Law - Does this mean I can now apply for my Canadian Citizenship?
An Act to Amend the Citizenship Act received Royal Assent on June 19, 2017. Therefore, you may be able to apply for Canadian Citizenship in the fall of 2017, if you meet the new requirements. Please note that not all changes to the Act will come into effect immediately. Some changes will be launched in the fall of 2017 with the remaining changes to come into force in early 2018.
I have been accepted into a Canadian University. How long does it take for my Study Permit application to be processed?
Generally, a foreign national who intends to study in Canada for 6 months or less, does not require a study permit. If the period of study is for more than 6 months, you will require a Canadian study permit. If you are studying for 6 months or less, you will require a study permit if you decide to extend your studies beyond 6 months or start another program after the initial studies have ended. If a study permit is issued to you as a full time international student and you meet specific criteria, you may be allowed to work on or off campus for up to 20 hours per week without having to apply for an additional work permit.


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