Tweedledum and Tweedledee

Jeremy Corbyn and Donald Trump

Two unlikely politicians have recently risen to prominence on both sides of the Atlantic.  Although one is on the extreme left and the other on the far right, both show uncanny similarities in the way they have risen to prominence and in their popular appeal.  Jeremy Corbyn was elected as Leader of the U.K. Labour Party against all odds, and Donald Trump has unbelievably become the Republican Party’s de facto candidate for President of the United States.  The mind boggles at the prospect of two of the free-world’s major economic and political powers being led by such implausible characters, but it could come to pass.

Their appeal appears to lie in populism, a simplistic view of world affairs and a promise to ‘do things differently’ by divorcing themselves from the governing establishment that has dominated politics in both countries for decades.  Of course, their differences in policy and style are enormous as one would expect of politicians at opposite ends of the political spectrum.  Trump would abolish Obamacare and abhors any state intervention, while Corbyn is a full-blown socialist who would re-nationalize major industries; Trump wants to ban Muslims from entering the USA while Corbyn embraces Hamas and Hezbollah; Trump believes in a USA respected world-wide for its military might, but Corbyn would scrap Trident and has talked about withdrawing from NATO; Corbyn’s open-door policy on immigrants contrasts with Trump’s intention to keep them out by building a wall along the US-Mexico border; Trump is the archetypical proud American who wants to “make America great again”, but republican Corbyn seems almost embarrassed by any display of patriotism; Trump is brash, outrageous and rude compared with the outwardly reserved and polite Corbyn who nevertheless has a barely concealed streak of ruthlessness; with his dyed blonde hair brushed forward and his comfortable and jovial presence on TV, Trump seems more of a politically-incorrect comedian than a politician, while Corbyn’s neatly trimmed beard and often scruffy manner of dress suggests a college lecturer rather than the leader of a party.  A surprising area of agreement, however, is that they both have favourable leanings towards Vladimir Putin and his policies.

How then, have two such extreme individuals gained prominence and positions of potential power despite the strong opposition of their respective parties’ mainstream politicians?  The answer seems to lie in support from the grassroots membership of the parties they represent − not the gun-lobby, the National Rifle Association or members of the Tea Party wing of the Republican Party who are Trump’s natural allies, nor the Marxist trade union leaders, militant socialists or radical students who would support Corbyn under any circumstances, but rather the rank and file Republicans from ‘Middle America’ and ordinary members of the Labour Party many of whom joined in order to elect Corbyn.   Thus many of Trump’s supporters are the smalltown blue-collar workers and rural folk, “the silent majority” as Richard Nixon called them, who haven’t travelled outside the USA, are worried about immigration and terrorism, are concerned that globalization and distant wars are damaging America’s economy and stature in the world, and who sense a decline in their way of life.  They are uneasy about the future and are beginning to wonder if they live in ‘God’s country’ after all!  On the other hand, Corbyn has garnered much support from relatively well-off middle-class members of the Labour Party − academics, media types, ‘luvvies’ in the arts world, educators and the like − who are burdened with a guilt-complex about Britain’s imperial past and who are disillusioned with the previous ‘New Labour’ governments led by Tony Blair and Gordon Brown, which in their view encouraged all the excesses of free-market capitalism that resulted in obscene salaries for bankers and financiers and a general widening of the wealth gap between rich and poor.  Among them are the so-called ‘champagne socialists’ and ‘chattering classes’, characterized by those who express deep concern about social injustice, the under-privileged and poverty while enjoying and protecting their own comfortable lifestyles.

My personal hope is that neither Tweedledum nor Tweedledee will ever be elected to positions of power.  It is a terrifying thought that a President Donald Trump would be just an arm’s length away from pressing the nuclear button, not against Russia but possibly as a reaction to cheeky provocations from North Korea or as an offensive attack on a territory controlled by Islamic State.  Moreover, America’s withdrawal from free-trade agreements and its retreat under Trump into isolationism and protectionism would have a far-reaching, damaging effect on the global economy.  The election of a well-meaning but inexperienced (and some would say naïve) Jeremy Corbyn as Prime Minister would, I believe, have equally damaging consequences for the British economy, returning it to the bad old days when dictatorial union leaders wielded a disproportionate amount of power that greatly contributed to the demise of the manufacturing industries (as Hugh Scanlon, one of the most militant union leaders at the time, later admitted).  I fear also that a Corbyn government would surrender its responsibilities in helping to maintain the security and defence of the free world, and would try to seek an accommodation with fanatic terrorist groups that have no interest in compromise.  There is a danger, in my opinion, that Great Britain would be reduced to an insignificant, weak and economically depressed country that would no longer be regarded seriously by its international partners.

With any luck we shall never know whether or not these depressing predictions make any sense because I am optimistic that Hillary will be elected President and there is always the possibility that the ineffective Corbyn will be replaced as leader of the Labour Party before the next general election.  We’ll see what happens.

John Weaver
May, 2016

Arithmetic

Counting and arithmetic are a child’s first introduction to mathematics and would be regarded by those who go on to study algebra, geometry, trigonometry and perhaps calculus at high school as the most elementary branch of mathematics.  Paradoxically, however, advanced arithmetic or number theory as it is usually called, is one of the most complex and difficult areas of mathematical research.  It includes still unsolved problems that can be stated in simple-to-understand language and which appear to be true by trial and error, but which are fiendishly difficult to prove.

One famous example, of course, was Fermat’s Last Theorem, first postulated in 1637, but not proved until over 3 centuries later by Andrew Wiles. The theorem states that there are no positive integers x, y and z satisfying xn + yn = zn  for integers n > 2 (that the equation has solutions for n = 1 is obvious and there is also an infinity of solutions for n = 2, for example  32 + 42 = 52, 52 + 122 = 132, etc.).  Although no one could find a solution for n > 2 it took until 1995 to show that none existed.

Other examples abound.  It is easy to show that there are infinitely many prime numbers, but what about primes of the form 2p – 1, the first few of which are 3, 7, 31, 127, corresponding to p = 2, 3, 5, 7 respectively?  Whether or not there is an infinity of them is still an open question even though the continual discovery of ever larger prime numbers of this type suggests they must be infinite in number (presently the largest one discovered corresponds to  p = 74,207,281).   Likewise, it is not known if there are infinitely many perfect numbers (a perfect number such as 6 or 28 equals the sum of its divisors excluding the number itself) nor indeed if there are any odd perfect numbers.  Seemingly simple problems, but no solutions yet.

In the accompanying note Foundations of Arithmetic I mainly discuss the Fundamental Theorem of Arithmetic.  Although it is the basis of elementary arithmetic, it is related in its analytical form to the Riemann zeta function thereby showing how arithmetical problems can lead one straight into advanced analysis.  That to me, as a total non-expert, is one reason why number theory seems such a fascinating subject requiring very special mathematical insights.  I can understand why it is an area of research that has always attracted some of the world’s most gifted mathematicians.  Also included in the linked document are a couple of simple and familiar results, the first of which is an exercise in mathematical induction when two variables are involved, while the second provides a gentle introduction to modular arithmetic.  It is all thoroughly well-known material, of course, expressed here in my own style and notation.  The books The Higher Arithmetic (Hutchinson, 1952) by H. Davenport and An Introduction to the Theory of Numbers (Oxford, 1960) by G. H. Hardy and E. M. Wright were my principal sources of information.

A Tale of Two Cities – Halifax and Victoria

Background
Halifax Regional Municipality and Greater Victoria have much in common. They are both provincial capital cities, they are of similar size (Halifax 414,000, Victoria 359,000 according to official Statistics Canada estimates1 in 2014), they are respectively the home ports of the Royal Canadian Navy’s Atlantic and Pacific fleets, within their regions they are relatively old cities founded by the British (Halifax in 1749, Victoria in 1843), and they both include suburban and rural communities in addition to their urban cores.

But there are significant differences as well. Halifax is not only the capital of Nova Scotia and the largest city in the province but is also the leading commercial centre and transportation hub for the maritime provinces. Victoria, on the other hand, is overshadowed by its younger but very much larger neighbour Vancouver which is not only the dominant city in British Columbia and the third largest metropolitan area in Canada but is also the location of the largest port and the second busiest airport in the entire country.

An even more striking difference between Halifax and Victoria, however, is how the two regional cities are governed. Greater Victoria comprises 13 separate municipalities each with their own mayors and councils (91 of them!), and 3 electoral areas, all overseen by a Capital Regional District (CRD) with limited authority and responsibilities, and whose Board members are appointed or elected by the various municipalities and districts themselves. The Halifax Regional Municipality is a single entity having been created in 1996 through an amalgamation of Halifax City, Dartmouth, Bedford and the County of Halifax. It is governed by the Halifax Regional Council (HRC) of 16 councillors and an elected mayor. Local matters are considered by three community councils, composed of several councillors from the HRC, which make recommendations to the Regional Council. This is the exact opposite of the situation in Victoria where it is the CRD that is subservient to the individual councils rather than the other way round. Board members of the CRD are accountable not to the region but to their municipalities and can, through their councils, block any recommendation from the CRD that would benefit the region as a whole but which they perceive as unfavourable to their own small municipality (and their own prospect of re-election!).

Whereas the Government of Nova Scotia, concerned that the disunity of its largest metropolitan area was hindering its economic development, decided to commission a study of governance in the Halifax region and subsequently introduced an Act to Incorporate in 1995, the present Government of British Columbia prefers a “hands-off” approach to municipal affairs and anyway appears to be totally disinterested in Victoria as it focusses almost exclusively on Vancouver where its support is based.

Andrew Sancton, a well-known expert on municipal amalgamations, states that the Halifax Act was largely the pet project of the Liberal Premier, John Savage, who as mayor of Dartmouth had, ironically, previously opposed amalgamation when it was first mooted by the former Progressive Conservative Premier Donald Cameron. His motives apparently were to reduce public spending and foster economic development, but Sancton claims there was little public interest in the matter and certainly no pressure from external sources for him to pursue this course of action, or in Sancton’s own words2:

“It is extremely difficult to argue that there were strong societal forces urging Canadian provincial governments in the 1990s to implement sweeping municipal amalgamations in major metropolitan areas. In Halifax, the Royal Commission on Education, Public Services, and Provincial-Municipal Relations called for a single municipality as early as 1974. In 1992 the provincially appointed Task Force on Local Government arrived at a similar conclusion. In neither of these cases was there great public interest in the issue. … Such policies were brought in with little or no thought by provincial premiers who acted as they did in response to the particular political circumstances in which they found themselves. They made little or no effort to mobilize consent for these policies beyond a small group of cabinet ministers who in turn helped control obedient caucuses.”

Circumstances in Victoria couldn’t be more different.  The amalgamation question has been alive and debated for decades:  numerous public discussions have taken place; several well-attended forums featuring panels of experts and distinguished guest speakers have been organized; citizens’ groups seeking an independent study of local governance have been formed (Amalgamation Yes, Grumpy Taxpayer$ of Greater Victoria, Greatest Greater Victoria Conversation); countless articles and letters on the topic have been published in the local press3; radio and TV interviews on amalgamation are a regular occurrence; and so on. In 2014 an Angus Reid poll determined that 84% of residents in the Capital Region favoured some sort of amalgamation with 50% “strongly in favour”.  An overwhelming majority of 89% supported a non-binding referendum on the issue, and 80% expressed support for an independent study and cost-benefit analysis of amalgamation in the region, extraordinarily high figures for a poll of this nature.

Local councils and mayors have traditionally been opposed to any change in the present governmental structure which is not surprising since many of them would be losing their positions in a unified region. Possibly influenced by the poll results, however, and reacting to pressure from various business, professional and citizens’ organizations, eight of the thirteen municipalities agreed to put some sort of question on the ballot when the municipal elections were held in November, 2014, asking if voters favoured a non-binding, independent review of governance in the region.  It was perhaps a reflection of the disunity in the region that there was no agreement on a simple, common question; rather each municipality phrased the question to suit their own purposes – one was so convoluted that the result could be open to various interpretations, another was worded virtually to invite a ‘No’ vote.  Five municipalities (fairly small ones) deemed the topic of no interest to their residents and declined to place any question on the ballot.  Despite this lack of enthusiasm by local politicians, the public responded with a higher than average turnout in those municipalities with a question on the ballot and a resounding 75% vote in favour of a review of local governance.

It is tempting to ask if Professor Sancton would regard the case for amalgamation in Greater Victoria different from and more compelling than that in Halifax given that it has grown out of a grassroots campaign rather than a top-down imposition by the provincial government.

Financial Considerations
The first thing opponents of the Halifax amalgamation will point out is that the promised savings were never realized in the short term.  They will mention the fact that salaries rose to the level of the best paid equivalent positions among the pre-amalgamation municipalities and that economies of scale were a dream never realized. According to Robert Bish4 the cost of implementing the amalgamation alone was four times the original estimate and he further stated that:

“It is not yet apparent that any cost savings will result.  From 1996 to 2000, user charges increased significantly and average residential property taxes rose by about 10% in urban areas and by as much as 30% in suburban and rural areas.”

All that was some time ago, of course, and although proponents of amalgamation are primarily interested in delivering a more efficient, more equitable and less complicated model of governance rather than in any potential cost savings that may result, it is nevertheless interesting to compare the current costs of government in the two regions, Halifax and Victoria, and their economic development.    The Executive Summary in Halifax’s proposed operating budget5 for 2015/16 begins with the statement:

“As a municipality, Halifax is in a strong, healthy financial position. The long term financial position of the municipality is generally sustainable as evidenced by its debt position.  In line with the long-term trend, the Tax Supported Debt of the Municipality continues to steadily decline.  Debt had peaked in 1998-99 at nearly $350m but at the end of 2015-16 is expected to total $256.3m, a decline of nearly $100m, more than 25%.  This has been achieved despite a substantial increase in the population and higher demands for capital expenditures.”

On this same theme, Halifax Councillor, Reg Rankin, was quoted in the February 4th, 2016, issue of ‘The Coast’, a Halifax newspaper, as stating

“Today, this is the equivalent of paying off two-thirds of our mortgage. We’re so pleased it has been accomplished over a generation, a generation I guess since amalgamation.”

Whatever Professor Bish’s post-amalgamation prognosis in 2001 might have been, it is apparent that in 2016 the Halifax Regional Municipality is in a strong and still-improving financial position.

Further evidence6 of its economic health is provided by its admirable economic development since 2005 when it was placed 15th out of Canada’s 28 large city regions in terms of GDP growth. It rose to 8th place in 2014 and is predicted to be 1st in 2015, while over the same period Greater Victoria has fallen from 4th to 27th position in the same table.  One cannot simply attribute Halifax’s remarkable economic growth to amalgamation, of course, but it certainly hasn’t hampered it!  Meanwhile the Greater Victoria Development Agency has recently identified the lack of collaboration and common goals among the region’s multiple jurisdictions as a major reason for its sluggish performance and lack of success in attracting available funding.  It proposed formation of a South Vancouver Island Development Association, funded by all constituent municipalities, to rectify the situation.  The proposal had to be ratified (or not!) by all 13 councils, of course, an obstacle and delay that would have been entirely unnecessary if Greater Victoria were a unified city-region.  In the end 11 municipalities and the Songhees First Nation signed the agreement with only two small semi-rural municipalities, Metchosin and Sooke, opting out.

A detailed comparison of revenues and expenditures in the two metropolitan areas is not a simple task that requires little more than a straightforward perusal of their operating budgets.  The figures for Halifax are readily extracted from the published Consolidated Financial Statements for the Regional Municipality that are available on the internet7, but the corresponding amounts for Greater Victoria are spread over 14 different financial statements which are not fully consistent in the way the data are presented.  A further complication arises because Halifax shows totals for the fiscal year ending on 31st March while the Greater Victoria municipalities use the calendar year for their financial statements.

There is also the “apples and pears” problem of trying to ensure that comparisons are like with like, since some services listed as part of the operating budget in one region may be treated differently in the other.  One obvious example is the cost of transit.  Halifax Regional Transit is a responsibility of the municipality whereas the Victoria Regional Transit Commission is a branch of BC Transit, which is partially financed by the Provincial Government.  In 2014-15, for example, the total cost of operating transit in Victoria was $120.5 million to which local municipalities contributed only $28.8 million from their property taxes8.  In the same fiscal year, expenditure for Halifax Transit was $110.9 million with property tax revenues contributing $76.3 million9, a far higher proportion than in Victoria.  The three school districts in the Capital Region receive funding from the Ministry of Education which in turn receives municipal taxes collected on its behalf by the constituent municipalities themselves.  The financial statement for Halifax clearly defines a single line item for ‘Educational Services’ but it is not at all clear in Greater Victoria what proportion of the Ministry’s grants to the school districts are funded by the various amounts collected by the municipalities. ‘Amortization’ is another significant entry on all the financial statements; these recorded amounts will depend on a number of factors that require detailed examination beyond my expertise.

There may be other examples of inappropriate comparisons that I am not aware of and, in any case, the different conditions existing in two separate provinces make meaningful interpretations of results somewhat tricky at best.  Harsher winters in Nova Scotia, for example, mean heavier road maintenance and snow removal costs, and the sheer size of Halifax Regional Municipality is a factor not reflected in population figures alone.  It covers a huge land area of 5490 km2 which is nine times larger than the area of Greater Victoria (696 km2) and even over twice that of the entire CRD (2341 km2) which includes the largely undeveloped and sparsely populated Juan de Fuca Electoral Area.  Halifax region hugs the coast for a linear distance of roughly 150 km and extends about 50 km inland. The CRD would have to expand as far as Nanaimo in order to encompass a comparable area.  No-one on Southern Vancouver Island is advocating an amalgamation on that scale.

Thus it soon became apparent that an unravelling of all the intricacies of these revenues and expenditures were best left to a qualified financial auditor.  Two parts of the financial statements, however, which can be retrieved relatively unambiguously, appear to lend themselves to a direct comparison.  They are wages and salaries, and protective services.  It is also a fairly simple exercise to compare the grand totals of revenues and expenses (slightly modified to account for differences arising from some incompatible ways of reporting revenues and expenses in the two regions).  The financial statements in these three areas are compared in the following paragraphs.

Wages and salaries.  Wages and salaries, including benefits, are stated in the first line of expenses in the Table of Segmented Information included in one of the ‘Notes to Financial Statements’ in the audited Consolidated Financial Statements for each municipality. They are shown here in Table 1 which suggests that Greater Victoria residents pay about $100 more per person for the salary and wages of mayors, councillors and municipal staff than the residents of Halifax Regional Municipality.  With 13 councils plus the CRD all employing their own administrative officers and staff this is perhaps hardly surprising.  Halifax’s one mayor receives a salary of $168,449 while the 13 mayors of Greater Victoria are paid a combined total of roughly $450,000.  All available evidence suggests that the cost of wages and salaries would be reduced in an amalgamated region.


Table 1

Expenditures on Salaries and Wages

(Figures in millions of dollars except last column)
Municipality 2013/14 2014/15 Average Per Capita
Halifax 324.0 343.5 333.8 $806
2013 2014
CRD 49.5 51.8
Victoria 104.1 107.3
Saanich 87.9 90.7
Esquimalt 12.9 13.5
Oak Bay 19.8 19.6
Central Saanich 10.5 11.0
North Saanich 5.2 5.4
Sidney 6.6 6.8
View Royal 4.1 4.3
Colwood 6.7 7.0
Langford 8.1 9.3
Metchosin 1.0 1.1
Highlands 0.6 0.6
Sooke 3.0 3.1
Total Victoria 320.0 331.5 325.8 $908

Protective services.   The police, fire and emergency preparedness services provided for citizens in all municipalities across Canada must be fairly standard.  Thus one would expect the cost per person for such services to be fairly uniform in communities of similar size.  “Protection services” is also a separate category of revenues and expenses on financial statements so a direct comparison between regions of similar size is fairly straightforward. There are, of course, significant differences between Greater Victoria and Halifax in how the delivery of such services is organized.  The Fire and Emergency Services Organization Chart for Halifax shows one Fire Chief supported by a Deputy Chief of Operations, a Deputy Chief of Operations Support, and a Manager of the Emergency Management Office.  Every municipality in Greater Victoria has its own Fire Chief and fire department, some of which are run on a volunteer basis, as are some of the rural components of the Halifax Regional Fire Department. The three former police forces in Halifax, Dartmouth and Bedford have been amalgamated into one Halifax Regional Police department responsible for policing those three former municipalities along with some neighbouring districts.  Policing in the rural areas is contracted out to the RCMP.  Greater Victoria, by contrast, maintains four separate departments in Victoria/Esquimalt, Saanich, Oak Bay and Central Saanich each with their own Police Chiefs and with markedly different caseloads per officer.  Two RCMP detachments, one based in Langford and the other in Sidney, serve the Westshore and north Peninsula respectively.  Small detachments are also present on Salt Spring and Pender islands which fall within the jurisdiction of the CRD.  The Senior Manager of the CRD’s protective services is also the Emergency Manager for the entire region.Table 2 shows compares the cost of protective services in the two regions.  The per capita expense in Greater Victoria is slightly greater ($20) than in the Halifax Regional Municipality.

Table 2
Expenditures on Protective Services

(Figures in millions of dollars except last column)
Municipality 2013/14 2014/15 Average Per Capita
Halifax 192.1 203.0 197.6 $477
2013 2014
CRD 8.5 8.7
Victoria 64.2 65.9
Saanich 47.2 50.3
Esquimalt 11.3 11.9
Oak Bay 8.9 8.6
Central Saanich 6.9 7.3
North Saanich 2.7 2.8
Sidney 4.0 4.1
View Royal 3.0 3.1
Colwood 5.2 5.5
Langford 8.6 10.1
Metchosin 0.7 0.7
Highlands 0.4 0.4
Sooke 3.0 3.2
Total Victoria 174.6 182.6 178.6 $497

Total revenues and expenditures.   The grand totals of revenues and expenditures in each jurisdiction are prominently displayed on all financial reports.  While the figures themselves provide a simple overview of how much tax and other revenue is generated in the two regions and what their overall operating costs are, they are not directly comparable in a meaningful way because of the “apples and pears” reasons given earlier.  Their interpretation is further complicated by the different formats used in reporting year-end financial statements in different municipalities.  For example, Sooke clearly identifies taxes received as “Net taxes available for municipal purposes” and in an explanatory note lists the actual property taxes collected less taxes levied on behalf of schools, CRD, hospitals, Municipal Finance Authority, BC Assessment and BC Transit.   North Saanich and Sidney also state that taxes shown under revenues are for municipal purposes, but others simply state “Taxes”.  I have assumed these are all net taxes since there are no payments to the various outside agencies listed under expenses and proportionally the amounts seem consistent with those for municipalities reporting net taxes.  Payments to the CRD are included under revenues in the CRD’s Financial Statement of course. Halifax also collects taxes for its school system but records the amount as a separate item in Revenue under Educational Services.  The identical amount is then shown under Expenses as a lump sum payment to Educational Services.  The funds simply flow through the Financial Statement without having any effect on the overall surplus or deficit.  In Greater Victoria such payments to outside agencies do not appear at all.  Thus for a fair comparison the amounts appearing under Educational Services have been omitted from the totals for Halifax in Tables 3 and 4 which show the total revenues and expenses for Halifax alongside the corresponding totals for the combined Capital Region.  (An even fairer comparison would have resulted if the taxes for transit in Halifax and its net operational expenses had also been removed from the totals because transit in Greater Victoria is run by BC Transit, not by the CRD or individual municipalities.)  Note that the total of expenses in the tables is not a measure of property taxes levied in the two regions. It excludes all the taxes collected on behalf of other agencies in Greater Victoria and those for educational services in Halifax.

Table 3
Revenues

(Figures in millions of dollars except last column)
Municipality 2013/14 2014/15 Average Per Capita
Halifax 778.9 815.0 797.0 $1925
2013 2014
CRD 181.8 191.6
Victoria 209.5 221.2
Saanich 189.1 187.5
Esquimalt 31.1 33.3
Oak Bay 34.0 36.7
Central Saanich 25.1 25.2
North Saanich 17.5 17.9
Sidney 18.8 20.1
View Royal 18.0 12.5
Colwood 19.7 19.4
Langford 55.1 51.1
Metchosin 5.3 4.8
Highlands 3.8 2.6
Sooke 13.6 16.2
Total Victoria 822.4 840.1 831.3 $2316

Table 4
Expenses

(Figures in millions of dollars except last column)
Municipality 2013/14 2014/15 Average Per Capita
Halifax 733.0 779.1 756.1 $1826
2013 2014
CRD 147.6 148.7
Victoria 173.9 174.0
Saanich 151.1 164.4
Esquimalt 29.5 30.2
Oak Bay 32.2 32.5
Central Saanich 25.8 25.2
North Saanich 15.1 15.9
Sidney 17.9 17.9
View Royal 12.1 12.4
Colwood 17.7 18.5
Langford 53.0 39.3
Metchosin 4.4 4.6
Highlands 2.7 2.8
Sooke 11.6 12.0
Total Victoria 694.6 698.4 696.5 $1940

What can we conclude from this discussion?  First, I think one can immediately put to bed the alarmist claims by some politicians that amalgamation would dramatically increase the costs of local government in the Victoria region.  The tabulated data, unprofessional as they are, indicate that after amalgamation Halifax has possibly lower rather than higher operating costs per person than the combined costs of the Capital Region’s 13 separate municipalities.  Indeed, they suggest that in certain areas there are probably modest savings to be gained through amalgamation of a mid-size city-region with a population of about 400,000.

Other Factors
Potential fiscal savings are not the main reason proponents support amalgamation, however, a point that defenders of the status quo seldom recognize. There are more compelling grounds of a wider nature admirably summarized by Rudiger Ahrend et al.10 who state in their abstract: “A city’s metropolitan governance structure has a critical influence on the quality of life and economic outcomes of its inhabitants.” … “Administrative fragmentation, which complicates policy coordination across a city, has a negative effect on individual productivity. This finding, combined with benefits from good governance such as improved transport and lower pollution levels, highlights the importance of well-designed metropolitan authorities.” Referring to the Organisation for Economic Co-operation and Development (OECD), the authors go on to say: “The OECD Metropolitan database defines ‘functional urban areas’ across the OECD on the basis of a common method that relies on settlement patterns and commuting flows rather than administrative borders.” … “A large number of municipalities in metropolitan areas can complicate policy coordination among local governments. A potential solution to this coordination problem could be the amalgamation of municipalities within a metropolitan area. ”On this definition Greater Victoria is certainly a single functional urban area as anyone who uses the Trans-Canada or Pat Bay highways at commuting time knows. So how do Halifax and Greater Victoria compare when measured against those factors related to economic well-being and quality of life referred to by Rudiger Ahrend et al.?  Of course, with its scenic setting, mild climate and general ambience Victoria has a natural built-in advantage with regard to quality of life, and it is fortunate to have a well-educated population including some, especially among those who have retired to the region, who are quite wealthy and willing to support arts and culture.  But there are other factors which should contribute to the economic success and efficiency in the two regions which are discussed below.

Policing.  With an amalgamated regional police force and one emergency call centre for its core districts it is inconceivable that Halifax could experience the shocking delay in responding to a frantic 911 call that occurred in Oak Bay some years ago.  The address was on the artificial border between Victoria and Oak Bay and the resulting confusion eventually involved three different police forces. Six hours later police discovered five bodies inside the house in a murder-suicide crime.  This was a terrible failure due to lack of communication between separate police forces.  Attempts to create integrated units for Greater Victoria have only had limited success with municipalities joining and subsequently withdrawing because of costs.  Kash Heed, a former BC solicitor general (and a former police chief in West Vancouver) is quoted as saying integration is a failed policy that fuels conflict rather than co-operation and that integration is a “band aid solution to a gaping wound”.  Another problem is the enormous difference in cost per capita among the municipal police departments within Greater Victoria, $455 for Victoria, $263 for Saanich, $257 for Oak Bay and $245 for Central Saanich in 2013.  This is because VicPD is responsible for policing the downtown core where much of the crime, drug-dealing and homelessness exists.  Ironically over 50% of downtown crime is committed by residents of other municipalities in the region. Halifax is well ahead of Greater Victoria in policing.

Influence in Ottawa.  Halifax is a member of the 21-strong Caucus of Big City Mayors which meets two or three times a year, sometimes with leaders of government, business, labour and the economy, to plan strategies for dealing with infrastructure renewal, homelessness, social housing, sustainability, and related issues.  Their most recent meeting in early 2016 was with Justin Trudeau, the newly-elected Prime Minister.  With a population of only 80,000 Victoria doesn’t qualify for membership of the Caucus but it is certainly burdened with the “big city” issues listed above.  It is frustrating to see smaller (but unified) cities such as Regina, Saskatoon, St. John’s, Windsor,  Gatineau, Kitchener, all having the ear of the Prime Minister while Victoria is excluded simply because it is a constricted urban core surrounded by other independent suburban municipalities.  Unfortunately, unless the region is represented by a single mayor it will be forever excluded from this influential body.  Halifax is poised to take advantage of funding made available to big cities while Victoria, Saanich, Langford etc. will compete with each other for the scraps left over.

Payments in lieu of taxes. Both Halifax and Greater Victoria receive substantial payments in lieu of taxes from the Federal Government for the Atlantic and Pacific Naval Bases and Dockyards respectively.  The difference is that while the funds going to Halifax benefit the whole region, in Greater Victoria it is the municipality of Esquimalt (population 16,800) that receives the entire payment representing over 40% of its tax revenue, even though many of the naval and dockyard personnel reside in other municipalities.  The general taxpayer in Halifax gets the fairer deal.

Sewage treatment.  Although scientific and medical opinion is firmly of the view that secondary sewage treatment in Victoria is unnecessary and a complete waste of money, the fact remains that it has been mandated by the Provincial Government.  The CRD took charge of the project and the history of the sorry saga that followed exemplifies all that is wrong with that body and with governance of the Capital Region.  After ten years of debate, planning, public consultation and engineering reports, and an expenditure of nearly $100 million there is still no site selected for the treatment plant(s) or a decision on whether there will be one, two or more plants.  The estimate of the overall cost now exceeds $1 billion. At one time a site had been selected but Esquimalt council refused to rezone the additional land needed to accommodate the plant.  One small municipality’s action caused the collapse of the entire project.  In Halifax, where treatment is most definitely needed (its effluent empties into the enclosed, sheltered waters of Halifax harbour) construction of treatment plants didn’t begin until after amalgamation because as separate cities, Halifax and Dartmouth could not reach agreement on the design of the plants.  Sewage treatment was completed in 2011 at less than a third of the estimated cost for Victoria.  And the moral of the story?  It took amalgamation to get things done.

Arts and recreation. Victoria had the foresight and good fortune to retain and restore two lovely Edwardian theatres, the 1416-seat Royal, home to Pacific Opera Victoria and the Victoria Symphony, and the 772-seat McPherson Playhouse used for plays, concerts and amateur productions.  Although they serve regional audiences and beyond, Victoria is the sole municipal contributor to the McPherson and only three municipalities (Victoria, Saanich and Oak Bay) contribute to the Royal Theatre.  (Professional theatre is also staged at the Belfry Theatre, a converted church building, and the Roxy Theatre, while the 800-seat Alix Goolden Hall and the 1228-seat University of Victoria Auditorium are alternative venues for musical concerts.  These are independently run facilities, however.)  Despite this rich choice of venues for Victoria’s vibrant cultural scene, what is lacking for a capital city is a modern performing arts centre with a large stage suitable for ballet and major touring productions, and acoustics that befit a fine symphony orchestra. Given the reluctance of other municipalities to provide funds for theatres located in downtown Victoria, it is not surprising that after decades of informal campaigns, studies and proposals there is no sign of any progress having been made towards building such a facility.  By contrast, Halifax is not nearly as well provided with performance spaces.  Its main professional theatre is the Neptune which seats 479 playgoers.  For larger productions and symphony concerts it uses the 1023-seat Rebecca Cohn Auditorium on the Dalhousie University campus. These are meagre facilities compared with Victoria, but the Halifax Regional Council does have a new performing arts centre officially on its agenda as one of the large projects in the concept phase.  With the council representing the whole of the region it is likely to be built while unofficial groups in Greater Victoria are still talking.  The same can be said for arenas.  Many of the municipalities in Greater Victoria have built their own community recreational centres which is excellent, but a city-region of 360,000 also needs a large arena suitable for a semi-professional hockey team, figure-skating and gymnastics competitions, trade shows and rock concerts.   Victoria’s tired old 4200-seat Memorial Arena was replaced in 2005 by a new 7000-seater courtesy of the City’s taxpayers alone. Halifax has an arena with seating for 10,600 spectators (12,000 for concerts) but is already considering building a bigger one with a capacity of over 15,000 spectators for hockey games.  If this comes to fruition the costs will be distributed equitably across the region.  One can only imagine what kind of arena Victoria could have had if it had been an amalgamated city when the aging Memorial Arena was replaced.

Garbage collection.  Each municipality in Greater Victoria has its own regulations and methods for garbage pick-up.  Whereas Victoria employs several men and one truck for residential services, Saanich, with a more automated system, needs only one man and a truck.  Langford doesn’t provide any garbage collection at all as a municipal service − residents have to make their own arrangements with a private company.  Some municipalities provide residents with two standard containers, one for general non-recyclable rubbish and the other for organic kitchen waste, others do not.  Halifax has a uniform method of garbage collection but it appears to be less advanced than some of those in Greater Victoria.

Neighbourhood representation. Halifax is divided into 16 Districts each electing one councillor.  The Districts are grouped into 3 Community Councils comprising the councillors elected in each District.  The Community Councils make recommendations to the Regional Council which has final authority.  Local public input and consultation is generally made at the Community Council level or, personally, to the District councillor.  In Victoria and Saanich, members of neighbourhood associations are ordinary residents interested in community activities, planning, development, and the general character of the areas in which they live. Board members of a neighbourhood association, who meet usually once a month, are all volunteers.  Each councillor is assigned to one or more neighbourhoods and attends their board meetings if possible. The associations may draw up neighbourhood plans, advise on land-use management and raise objections to unwanted developments, but they have no final authority which rests with the municipality.  It would be possible to copy the model of regional governance in Halifax by dividing the CRD into perhaps four community councils, say (i) East Victoria, Oak Bay, Gordon Head and Cadboro Bay, (ii) West Victoria, Esquimalt, View Royal and South Saanich, (iii) West Shore and Juan de Fuca, and (iv) Cordova Bay, Prospect Lake, Peninsula and Islands, with each “Community” sub-divided into three or four “Districts”. Some members of the neighbourhood associations are ambivalent about amalgamation as they feel it would relegate them to an even more remote position of influence from the ultimate decision-makers, but perhaps they could play a similar role in their District if the Halifax model were adopted.

Rural protection. Some opponents of amalgamation claim it would threaten protection of rural areas.  Halifax includes a large rural component and it remains intact.  Saanich, the largest of the Greater Victoria municipalities has managed to preserve its rural part despite the encroachment of suburban housing.  It is the smaller municipalities that are more worrying, as it is harder for them to resist the temptation of increasing their tax base when presented with attractive offers from developers.  Langford cleared forested areas, including a Garry Oak meadow, in order to develop its cluster of box stores.  Central Saanich, a mainly rural community, has allowed an ugly strip of light industry, warehouses and commercial businesses to grow along Keating Cross Road, and now a new proposal has emerged to build yet another shopping mall on the outskirts of Sidney, the last thing that compact community needs.  A single regional plan for the entire region would surely help to prevent such haphazard developments from happening.  In my opinion, amalgamation would provide better rather than less preservation of the rural areas in the Capital Region.

Emergency preparedness.  Halifax has one emergency plan and one emergency call centre. Victoria, situated in an earthquake zone, has 13 plans and 3 call centres.  The mind boggles at the sort of chaos that will develop if the ‘big one’ occurs with the present administrative structure still in place.

Infrastructure renewal, traffic and planning. Two bridge replacements demonstrate the unfair burdens placed on individual municipalities when infrastructure serving the entire region is renewed.  The Johnson Street bridge is on one of three routes connecting the West Shore to downtown and beyond.  Yet it lies entirely within Victoria which must therefore bear the full cost, even though it is Esquimalt which probably receives the most benefit in this case.  Much of the funding for the Craigflower Bridge upgrade came from the gas tax, but Saanich and View Royal, the municipalities at either end of the bridge (just – the Esquimalt border nudges up to the corner of the bridge on the View Royal side), each contributed $2.5 million.  Much of the traffic on this bridge, especially at commuting time, is to and from the naval dockyard in Esquimalt, which was not financially involved in this project.  Planning in the region is not well coordinated.  Langford’s box stores were planned by that city alone, yet they attract traffic from other parts of Greater Victoria causing congestion on the Trans Canada highway and on the road leading from the stores to the highway.  Langford is, however, anticipating the possibility of a reopening of the E & N railway for commuter trains between the West Shore and downtown while, ironically, Victoria has severed the rail link into the city centre because it couldn’t afford the additional cost of incorporating it into the new Johnson Street bridge.  There appears to be no coordinated plan in Saanich and Central Saanich to identify potential arterial routes through their municipalities to the airport and BC ferry terminal, with the result that all through traffic to those destinations is funnelled onto the already busy Pat Bay highway.  And surely it is only in Victoria that one encounters such bizarre situations as a plethora of building codes, designated bicycle lanes ending abruptly when reaching an invisible line between two municipalities, a default speed limit that suddenly changes when driving along what appears to be the same road at a constant speed, a suburban road where one side is like a country lane with no kerb or sidewalk while the other is a typical suburban street, a house that receives two partial tax bills because it happens to straddle two municipalities on an ordinary suburban street. All quite hilarious, but unfortunately true.  I know little about Halifax in these departments but I’m sure it has a regional plan, a standard building code, uniform traffic regulations and no invisible boundaries!

Conclusion
Amalgamation of the Halifax municipalities was not perfect.  Almost certainly it covered too large an area.  With its historical urban core centred on Halifax, Dartmouth and Bedford surrounded by vast tracts of rural land dotted with small communities, it is a marriage of barely compatible parts.  In such a large region some rural residents probably feel disconnected from the city and with their different priorities may perhaps disproportionately influence policy issues on such urban issues as libraries and transit.  Greater Victoria’s geography is rather different.  Only the small municipalities of Sooke, Metchosin and Highlands on its fringes are truly rural (and even they also serve as bedroom communities for daily commuters) while Central and North Saanich and parts of Saanich itself contain pockets of productive and protected farmland. It resembles a more cohesive community containing a mix of countryside, suburbia and urban core which geographically, at least, is probably even more suited to amalgamation than Halifax.Otherwise, however, amalgamation of the Halifax region appears to have been a great success.  The amalgamated city is thriving economically, it has reduced its debt in spectacular fashion and it punches well above its weight in national affairs.  I believe one would be hard-pressed to find local leaders, businesspeople, members of the arts community and government officials yearning for a return to its previous balkanized structure of governance.  Amalgamation has worked well for Halifax.  It should be held up as an example that Greater Victoria should emulate.

Amalgamation?  YES!

John Weaver
February, 2016

References
1Annual population estimates by census metropolitan areas, July 1, 2014, http://www.statcan.gc.ca/dailyquotidien/150211/t150211a001-eng.htm

2Andrew Sancton: Why Municipal Amalgamations? Halifax, Toronto, Montreal, in “Canada: The State of the Federation 2004, Municipal-Federal-Provincial Relations in Canada” (Robert Young & Christian Leuprecht editors), McGill University Press, 2006, pp 119-134.

3See http://www.amalgamationyes.ca/media–events.html for a selection of such articles and letters.

4Robert L. Bish, Local Government Amalgamations; Discredited 19th-Century Ideals Alive in the 21st Century, C. D. Howe Institute Commentary No: 150, March 2001, pp 35.

5Proposed Operating Budget 2015/16, Section A, Executive Summary,  https://www.halifax.ca/budget/documents/proposed_operating_book.pdf

6A New Regional Strategy and Model for Economic Development in South Vancouver Island, 2015 Report published by the Greater Victoria Development Agency Board, pp 46.

7Consolidated Financial Statementshttp://www.halifax.ca/finance/

8How Victoria Regional Transit is Funded,  http://bctransit.com/*/about/facts/victoria

9Proposed Operating Budget 2015/16, Section I3, Halifax Transit Operating Budget Overview, https://www.halifax.ca/budget/documents/proposed_operating_book.pdf

10 Rudiger Ahrend, Alexander C. Lembcke, Abel Schumann, Why metropolitan governance matters a lot more than you think, VOX, CEPR’s Policy Portal (Research based policy analysis and commentary from leading economists), 19 January 2016, pp 6: see http://www.voxeu.org/article/why-metropolitan-governance-matters

11Police resources in British Columbia, 2013, http://www2.gov.bc.ca/assets/gov/law-crime-and-justice/criminaljustice/police/publications/statistics/2013-police-resources.pdf

 

1 + 2 + 3 + … = – 1/12 and Related Results

The seemingly bizarre result in the title of this article has surprisingly found its way into some branches of advanced physics including string theory.  Its ‘proof ‘ is demonstrated in one of the Numberphile videos on YouTube where divergent series are manipulated in an easy-going manner, but the series can also be regarded in a more mathematically acceptable light as an expression of the Riemann zeta function ζ(z) evaluated at z = 1.

It occurred to me that it should also be possible to sum powers of the natural numbers using the same carefree operations on divergent series as in the Numberphile video.  In a discussion with the title Some Fun with Divergent Series, I first evaluate the sum of the squares of the natural numbers S2 and then, after a digression on Abel summation and the Riemann zeta function, I generalise the method to find the sum Sn of the series of natural numbers raised to the nth power.  Initially a recursion relation is found for a related quantity Tn  , the sum of powers of the natural numbers with alternating signs.  This leads directly to the formula for Sn and its connection with the well-known Bernoulli numbers.

After completing this article I discovered a WordPress blog by the eminent Australian/American mathematician Terry Tao. In one of his published articles there, entitled The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, he explores the summation of powers of the natural numbers in much greater detail and, of course, with considerably more insight than anything I could offer.  Nevertheless I have kept my approach posted here as it is a natural generalisation of the informal method shown in the Numberfile video referred to above.

Since I have been unable to find a way to embed a PDF file in a WordPress blog, the best I can do is to provide this link to Some Fun with Divergent Series.



The Monty Hall Problem

I had occasionally seen the TV programme ‘Let’s Make a Deal’ hosted by Monty Hall, but it wasn’t until I read The Curious Incident of the Dog in the Night-time, an acclaimed novel by Mark Haddon (2003), that I realized the TV show had sparked an interesting and controversial question in probability.  The main character who narrates the story in the book is a boy with Asperger’s syndrome and an unusual talent for mathematics.  Among his varied and unconventional musings he describes the Monty Hall problem in which a contestant was asked to choose one of three doors in the hope of winning a car which was hidden behind one of them.  After the contestant had picked a particular door Monty Hall would open one of the two other doors revealing a goat.  He then asked the contestant if he wanted to switch to the other unopened door or stick with his original choice.  The boy gives two proofs, one based on probability theory and the other on a flow-chart depicting all the possible outcomes, to show that the contestant should always switch because the probability that the car is behind the other unopened door is 2/3.  What surprised me, however, were the boy’s quotations of actual remarks made by PhD mathematicians claiming the solution to be wrong!  It is understandable that a first intuitive reaction would suggest that a contestant faced with only two closed doors, one of them concealing a car, had a 50% chance of winning whether he switched or not.  But in fact the Monty Hall problem is an example of Bayesian theory where posterior probabilities affected by new evidence can sometimes yield counter-intuitive results that have, apparently, confused even professional mathematicians on occasions.

Further investigation led me to an interesting blog by Allen Downey where this problem and others involving Bayes’ theorem are discussed.  Downey also refers to the Wikipedia article on the Monty Hall problem which explores in great detail many variants of the basic problem.  I find it remarkable that the strategy for winning a car in a simple TV game-show has given rise to such profound investigations in probability theory.

There are no new results in the following article.  I have simply developed well-known solutions in my own words in order to clarify my understanding of the theory.

The Monty Hall Problem

There are three doors.  Behind one of them is a car.  Behind the other two are goats.  The contestant is asked to pick a door, which he does.  The host then opens one of the other doors behind which there is a goat, after which he asks the contestant if he wishes to switch his pick to the other unopened door or stay with his original choice.  What should the contestant do?

Part 1
Let the doors be labelled A, B and C, and let the corresponding events that the car is behind these doors be A , B and C respectively.  Denote the probability that the car is behind door A by P(A) etc. and suppose the contestant chooses door A and the host opens door B to reveal a goat.  Clearly 𝑃(𝐴) = 1/3.  Hence we may infer that P(Ã) = 1 − 𝑃(𝐴) = 2/3 where Ã is the event ‘not A’ (the car is not behind door A).  Thus, since the events A, B and C are mutually exclusive (the car can be behind only one door), we have

2/3 = 𝑃(Ã) = 𝑃(𝐵 ∪ 𝐶) = 𝑃(𝐵) + 𝑃(𝐶).

 When the host opens door B to show there is no car there, the contestant knows that 𝑃(𝐵) = 0.  Hence from the equation above 𝑃(𝐶) = 2/3.

Thus we have P(A) = 1/3 and P(C) = 2/3 showing that the contestant should always switch his choice to the other unopened door.

Alternatively this result can be reached through Bayes’ Theorem, an approach that also prepares us for the more detailed investigation in Part 2.  Let 𝐸 denote the event of the host opening door B to reveal a goat. Then 𝑃(𝐸|𝐴) is the probability of him doing this knowing that the car is hidden behind door A.  Likewise, 𝑃(𝐸|𝐵) and 𝑃(𝐸|𝐶) are respectively the probabilities of the host opening door B when the car is behind doors B and C respectively.  It is obvious that 𝑃(𝐸|𝐵) = 0 (the car is there, not a goat!) and 𝑃(𝐸|𝐶) = 1 since he must open door B in this case (he cannot open C because the car is there, and door A, the contestant’s choice, always remains closed of course).  When the car is behind door A the host can randomly choose either door B or C as the one to open.  Thus 𝑃(𝐸|𝐴) = 1/2.  Bayes’ Theorem states that

                    𝑃(𝐴|𝐸) = 𝑃(𝐸|𝐴)𝑃(𝐴) / 𝑃(𝐸)

                                   = 𝑃(𝐸|𝐴)𝑃(𝐴) / [𝑃(𝐸|𝐴)𝑃(𝐴) + 𝑃(𝐸|𝐵)𝑃(𝐵) + 𝑃(𝐸|𝐶)𝑃(𝐶)].

The denominator follows because, as stated earlier, the events A, B and C are mutually exclusive.  Substituting the values we obtained above, we obtain

𝑃(𝐴|𝐸) = (1/2 · 1/3) / [1/2 · 1/3 + 0 · 1/3 + 1 · 1/3] = 1/6 / 1/2 = 1/3.

Similarly

𝑃(𝐶|𝐸) = 𝑃(𝐸|𝐶)𝑃(𝐶) / 𝑃(𝐸) = (1 · 1/3) / 1/2 = 2/3.

Either one of these results infers the other, of course, since 𝑃(𝐴 ∪ 𝐶) = 1.

Part 2
Part 1 describes the simple approach to the problem in which no account has been taken of the host’s strategy in opening the doors.  Obviously if the car should be behind the door A chosen by the contestant, then the host could open either door B or door C, but if it were behind door C then he can only open door B.  The point is that the host’s role in selecting which door to open has not been used in the solution so far.

The host may have a habit of preferentially opening door B, rather than door C, when the car is behind A, say 75% of the time.  But if he randomly chooses between doors B and C then the probability that he opens door B is 50%.  In general we can say 𝑃(𝐸|𝐴) = 𝑝  where 𝑝 varies from 0 to 1. When 𝑝 = 0 the host never opens door B to reveal a goat, presumably because he knows the organisers of the game always place the car behind door B.  At the other extreme 𝑝 = 1 he always opens B in the knowledge that the car is never placed behind that door.

Using Bayes’ Theorem again and following exactly the same arguments given in Part 1, we deduce

𝑃(𝐴|𝐸) = 𝑝 · 1/3 / [𝑝 · 1/3 + 0 · 1/3 + 1 · 1/3] = 𝑝/( 𝑝 + 1).

Suppose first that 𝑝 = 1/2, i.e. that the host randomly selects door B as the one to open.  Then 𝑃(𝐴|𝐸) = 1/3 which is the result obtained in Part 1.  If door B is opened by the host 75% of the time, then 𝑃(𝐴|𝐸) = 75/175 = 3/7 implying that the contestant should still switch his choice to door C where the probability of winning the car is 4/7 compared with 3/7 if he sticks with his original selection of door A.  When 𝑝 = 0 it is certain that the host will not open door B and is therefore forced to open door C.  Moreover 𝑃(𝐴|𝐸) = 0 in this case, so the contestant will definitely win by switching to the unopened door, namely B.  Finally, if 𝑝 = 1 the host must open B leaving the possibility that the car is behind either A or C, and indeed 𝑃(𝐴|𝐸) = 1/2 when 𝑝 = 1.

In general we have 0 ≤ 𝑃(𝐴|𝐸) ≤ 1/2 over the range of values 0 ≤ 𝑝 ≤ 1 so that it always pays the contestant to switch to the unopened door (except for 𝑝 = 1 when there is no advantage gained or lost by switching).

Morley’s Miracle

Although I have a background in mathematics, I confess to having never heard of ‘Morley’s Miracle’ until a few years ago when it was brought to my attention by a friend, a former high school mathematics teacher.  The ‘miracle’ is a theorem in elementary geometry stating that the trisectors of the angles of a triangle intersect in three points that form the vertices of an equilateral triangle. There are many solutions posted on the internet.  The one I came up with involves only elementary trigonometry and the Law of Sines.  It is perhaps slightly simpler than, although very similar to the first proof given here which uses both the Law of Sines and the Law of Cosines.

Click on this link to see my solution.

Dates of Easter, Pascha and Passover

Whenever the Eastern Orthodox Church celebrates Easter one or several weeks after the Western Easter, the inevitable questions arise as to why this is so.  Some of the responses I have heard over the years simply didn’t ring true and even on official-looking web sites I have seen statements that confuse rather than clarify the issue — such assertions as “the Eastern Church sets the date of Easter according to the actual astronomical full moon” (not true), “the Eastern Orthodox Church also applies the formula so that Easter always falls after Passover” (meaningless), “Orthodox Easter is always celebrated the Sunday after the Jewish Passover” (wrong),  “the rule set forth by the First Ecumenical Council … requires that Pascha must take place after the Jewish Passover in order to maintain the Biblical sequence of Christ’s Passion” (a redundant ‘rule’ on the Julian calendar for which there is no evidence).

In 2012 I wrote the following piece in an attempt to clarify the confusion about Easter dates in my own mind and I have passed it on to a few people who have asked for an explanation of how the dates are determined.  Before posting it here I have added a couple of new references including one to a very recent (March 2015) blog which supports much of what I have written.

Easter, Pascha and Passover

Most people are aware of the Julian and Gregorian calendars and know that most Orthodox churches use the former while the rest of Christendom follows the latter. They may not be as familiar, however, with the rules for determining the dates of Easter (or Pascha in the Orthodox Church) and why they are usually, but not always, different on the two calendars. Before delving into these differences let us remind ourselves of the historical origins of the two calendars.

In an attempt to coordinate the days of the year with the orbit of Earth about the sun, the Julian calendar was authorised by Julius Caesar in 46 BC as a replacement for the rather haphazard Roman calendar that had been used hitherto. The new calendar was based on the best advice available from astronomers at that time. It comprised a year of 365 days with the added proviso that every fourth year, what we now call a leap year, would include an additional day in order to keep the calendar in line with Earth’s orbital period and hence with the seasons as well. All went well for a few hundred years but by the 16th century it was clear that the equinoxes (when the sun is directly over the equator) were slipping backwards relative to the calendar. This was of particular concern to the Roman Church because the date of Easter was supposed to be determined by the vernal (or spring) equinox in the Northern hemisphere which was arriving ever earlier in the calendar year. In fact all Christian feast days were no longer synchronised with the seasons; Christmas was drifting towards spring and Easter was heading inexorably into the summer. In other words, it was realised that the actual solar year, the time it takes Earth to complete its orbit around the sun, was slightly shorter than the Julian calendar year introduced back in 46 BC.

Although this discrepancy was only a few minutes, over the centuries it had accumulated into several days. In order to correct this error, Pope Gregory XIII accepted a recommendation for calendar reform from a Jesuit astronomer, Christopher Clavius. It involved a one-time omission of ten days from the calendar to compensate for the excess of days already accumulated on the Julian calendar, and thenceforth a refinement in how leap years were to be determined – the extra day would no longer be included in years divisible by 100, except for those that were also divisible by 400. For example, the year 1900, although divisible by 4 and therefore considered a leap year in the Julian calendar, was not a leap year on the Gregorian calendar. On the other hand, the year 2000 was a leap year on both calendars because it is divisible by 400. Thus, on the Gregorian calendar Russian Orthodox Christmas was on the 6th of January in the 19th century, and will shift to the 8th of January after 2100, but it remained fixed on the 7th of January at the last millennium.

The new Gregorian calendar was implemented by Roman Catholic countries but its acceptance elsewhere was spasmodic, possibly because it was viewed with some suspicion in certain quarters as a papist innovation. Great Britain did not adopt the Gregorian calendar until 1752; Russia introduced it as a civil calendar after the 1918 revolution; and it was not until 1923 that Greece made the change by which time they had to remove 13 days from their calendar year. Despite its Roman Catholic origins, the Gregorian calendar is now recognised by most countries as the universal civil calendar.

The Anglican, Lutheran and other churches have also long since accepted the new calendar and based their liturgy on it, but the Orthodox Church has remained steadfastly resistant to change. At a synod held in Constantinople in 1923, however, a proposal for a ‘Revised Julian Calendar’ was introduced. Despite its name, the Revised Julian calendar is virtually the same as the Gregorian calendar, differing only by a small refinement in the way leap years are calculated. In fact, up to the year 2800, which will not be a leap year on the Revised Julian calendar, the two calendars are identical. It was accepted by some Orthodox churches, notably the Greek Church, but rejected by others, notably the Russian Church. That is why, for example, the Greeks celebrate Christmas on the 25th of December but the Russians on the 7th of January.

At the same synod a revised method for calculating dates of Pascha was also recommended. It was based on exactly the same principles originally promulgated by the 1st Council of Nicaea in the year 325, but expressed in a more precise form. Thus Pascha was stated to be the Sunday following the 24-hour day on the meridian passing through the Church of the Holy Sepulchre in Jerusalem during which the first full moon after the vernal equinox occurred, with the additional clarification that when the exact instant of full moon happened to fall on the same day as the instant of the equinox, the latter must precede the former (otherwise the next full moon would be the relevant ‘Paschal moon’). This would have removed the ambiguities present in the original definition but unfortunately there was no agreement among the various Orthodox churches to adopt this proposal and, with the exception of the Finnish and Estonian Orthodox churches (which have adopted the Gregorian calendar), they have continued to use the Julian calendar for determining the dates of Pascha.

Which brings us to the main theme of this discussion – how are the dates of Easter (Pascha) determined and what role, if any, does the date of the Jewish Passover play in their determination? In the Early Church, Easter was celebrated on various dates by different Christian communities. Some celebrated Easter on Passover itself whatever the day of the week, others on the Sunday following Passover, and yet others calculated an appropriate date for themselves. Moreover, those who were basing Easter on the rather erratic Jewish calendar at that time sometimes found that Passover occurred twice in the same year! The bishops assembled for the Council of Nicaea attempted (among the many other important decisions they made) to bring order to this chaos. Although precise minutes of their deliberations are apparently not recorded, the theme of their pronouncements on this topic emerges very clearly from a subsequent letter sent by Emperor Constantine to all those not present at the Council. First, all churches should celebrate Easter on the same day; second, this day should be a Sunday; and third (expressed by Constantine in very strong anti-Jewish rhetoric), Easter should be completely divorced from the Jewish calendar (see Documents from the First Council of Nicaea scanned from ‘The Seven Ecumenical Councils of the Undivided Church’, Vol. XIV, 1988, Henry R. Percival (ed.), which is also available on the Fordham University website).

The Council agreed that the date of Easter should be on the Sunday following the first full moon after the vernal equinox. Although this appears at first sight to be a unique date based solely on astronomical observations, it is in fact fraught with the ambiguities alluded to earlier. The vernal equinox is not always on the same day; it can occur on any of the five days from the 18th to the 22nd of March; the instant at which the full moon appears can have different dates at places on different longitudes. Even with the approximate time zones in modern usage, for example, a full moon at 12:30 a.m. on the 22nd of March in Jerusalem is observed in Rome at 11:30 p.m. on the 21st.

To overcome these difficulties, it was decided that for ecclesiastical purposes the vernal equinox would be deemed always to have occurred on the 20th of March, and tables of the ‘ecclesiastical full moon’ were drawn up which likewise specified dates of the full moon in any year to be used for calculating the date of Easter. Although the ecclesiastical full moons were only approximations to the astronomical full moons, they were remarkably accurate considering how long ago they were calculated. Finally, the Paschal full moon was defined as the ecclesiastical full moon after the 20th of March (the hypothetical equinox). Putting all this together, we can state an accurate definition of Easter and Pascha as follows:

Easter falls on the Sunday following the date of the Paschal full moon for that year

(R. W. Mallen, 2002, Easter Dating Method). Note that this definition faithfully follows the decisions taken by the 1st Nicene Council, is equally applicable in both the Eastern and Western churches, and is not dependent on the date of Passover. Many writers claim that the Council specified the 21st of March rather than the 20th as the fixed date for the vernal equinox, in which case the definition of the Paschal full moon should be modified to read ‘the ecclesiastical full moon on or after the 21st of March’.

The reason for the different dates of Easter and Pascha is now clear. Because the equinox was fixed to a calendar date rather than astronomical observations, the dates of Easter and Pascha started to diverge once the Western Church adopted the Gregorian calendar. The earliest possible date for Easter is the 22nd of March, in a year when the Paschal full moon falls on a Saturday that also happens to be the 21st of March. According to the Julian calendar, however, the 22nd of March is presently the 4th of April on the Gregorian (and civil) calendar. Thus Orthodox Pascha can never occur before the 4th of April and as the centuries pass, this date will gradually drift ever away from spring, from Passover and from Western Easter. After the year 2698, Easter and Pascha will never again coincide. Another less dramatic factor affecting the computed dates, which is nevertheless the reason why a one-week separation of Easter from Pascha is fairly common (as in 2012, for example), is that when the Gregorian calendar was adopted the tables of Paschal full moons were also corrected. The Orthodox churches have retained the original tables which means that the tabulated Orthodox Paschal full moons are now occurring a few days later than the corresponding Gregorian Paschal full moons.

So what have all these determinations of Easter dates to do with the Passover? In my view, absolutely nothing! None of the algorithms and tables used to calculate the dates of Easter and Pascha require information about the date of Passover. I have been unable to find any evidence at all to support the statements regularly made by many Orthodox priests and believers that the 1st Nicene Council demanded that Easter should always follow Passover. Rather the opposite; the intent of the Council was that Easter should in no way be linked to the Jewish calendar. Nicolas Ossorguine, an Orthodox theologian at St. Sergius Theological Institute in Paris writes:

The idea that the Christian Pascha must always be observed following the Hebrew Pesach was advanced by Byzantine canonists during the time preceding the introduction of the Gregorian calendar in the West (1583 AD). Apparently it was done in an attempt to discredit this forthcoming “Catholic” calendar reform.  

(See this translation by Alvian Smirensky in 2002 from the original article written in 1979). I think there is strong evidence to support this statement because the date of Pascha calculated on the Julian calendar will de facto always follow Passover. To state it as an additional requirement is as redundant as saying Easter must follow Valentine’s Day!  It always does. The reference to Passover can only have been introduced in a negative sense in order to refute the validity of the Gregorian calendar which does occasionally allow Easter to precede Passover. More recently, Archbishop Peter of the Orthodox Church in America wrote in the 1994 April/May edition of The Orthodox Church Newspaper:

There is among the Orthodox a very widespread belief that the Christian celebration of Easter must necessarily come after the Jewish Passover. This chronological order is considered imperative and bears a symbolic meaning, as it is believed to have been decreed by the First Ecumenical Council … Yet, not only is such a stipulation totally absent from the decision taken on the Paschal question at Nicea, but it is foreign and, in a sense, contrary to what was then decreed. 

And in an interesting blog posted in 2015, Fr. Andrew Damick (an Orthodox priest) labels the rule that Pascha has to be after Passover as Urban Legend Number 1 tracing its origins to a rather casual observation by a 12th century canonist.  It is remarkable that a rule based on such flimsy and unsubstantiated grounds can acquire the status of Church doctrine merely through constant repetition and unquestioning acceptance over many years.

I am certain that the immediate response from many Orthodox adherents will be “so what?”; Pascha must indeed always follow Passover in order to preserve the chronological order of events that took place 2000 years ago. Jesus rode into Jerusalem to celebrate the Passover and was crucified shortly thereafter. Consider, however, the dates of Easter and Pascha in 2013 when the vernal equinox occurred at 11:02 on the 20th of March universal time and the next full moon was at 09:27 on the 27th of March. Passover was on the 26th of March and Gregorian Easter fell on the 31st of March. It satisfied all the alleged criteria the Orthodox Church prescribes by occurring on the Sunday after the full moon after the vernal equinox and after Passover. Surely, then, 2013 must have been one of those years in which Easter and Pascha were coincident. But no, Pascha didn’t arrive until the 5th of May, a full 5 weeks later!  How this represents a more accurate representation of the historical chronology of the Resurrection relative to Passover escapes me. The reason for the delay is, of course, that the earliest Pascha can occur has slipped to the 4th of April as explained above and has nothing at all to do with the date of Passover. Indeed, if preservation of the order of the historical events is so paramount, why not revert to the practice of some early Christian groups by always celebrating Pascha on the Sunday following Passover? This would satisfy those who are adamant that Pascha must follow Passover, and would do so with historical accuracy according to the Jewish calendar, but that would then be completely counter to the directives of the 1st Nicene Council.

On a previous occasion when a similar situation to that in 2013 arose, an Orthodox friend justified it by maintaining that it wasn’t just Passover itself but the full Passover week that must precede Pascha. That explanation can be put to rest immediately by considering the relevant dates in the year 2014, when Passover week ran from the 15th to the 22nd of April and Easter and Pascha were celebrated together during that very same week, on the 20th of April.

A final argument in defence of the ‘Old Calendar’, which is sometimes used by devout Orthodox believers, is that it must be true because certain miraculous events such as the spontaneous combustion of the Holy Fire and the appearance of images in the sky always occur on Orthodox Pascha, not on the dates for Easter on the Gregorian calendar. For me, such claims merely diminish the credibility of the events rather than add authenticity to the Julian calendar. I cannot envisage a God that keeps track of man-made, ever-shifting and inaccurate dates that no longer represent the equinoxes in the Sun-Earth system that He created, in order to ensure that His miracles occur on the right day at a specified time. I recognise, however, that beliefs in such miraculous happenings are personal and emotional, and that no amount of rational persuasion will change them.

There are a number of web sites that provide automatic calculators of the dates of Easter, Pascha and Passover for any given year. One that is particularly user-friendly can be found here.  Interesting consequences arise if one looks forward several millennia. In the year 7010, some 5000 years from now, Passover will occur on the 19th of April, Easter on the 22nd of April and Orthodox Pascha on the 10th of June, 52 days after Passover! And while Orthodox churches remaining entirely on the Julian calendar will experience Christmas and Pascha slipping in tandem relative to the seasons, those that have adopted the Revised Julian calendar for fixed feast days but not for Pascha will find Pascha gradually closing in on Christmas as the former advances into summer and autumn while the latter remains fixed. Eventually calendar reform will have to be adopted, and I hope when that time comes all Christian churches will settle on a fixed date for Easter (Pascha), perhaps on the second Sunday in April.

John Weaver